Theorem funcsn2 ≪ | index | src | ≫ 

theorem funcsn2 (F: set) (a b: nat) {x: nat}:
  $ func F a (sn b) <-> F == \. x e. a, b $;
StepHypRefExpression
1 bitr4 
(func F a (sn b) <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b)) ->
  (F == \. x e. a, b <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b)) ->
  (func F a (sn b) <-> F == \. x e. a, b)
2 funcal 
func F a (sn b) <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b)
3 1, 2 ax_mp 
(F == \. x e. a, b <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b)) -> (func F a (sn b) <-> F == \. x e. a, b)
4 bitr4 
(F == \. x e. a, b <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x = b)) ->
  (isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b) <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x = b)) ->
  (F == \. x e. a, b <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b))
5 eqsrlam 
F == \. x e. a, b <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x = b)
6 4, 5 ax_mp 
(isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b) <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x = b)) ->
  (F == \. x e. a, b <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b))
7 elsn 
F @ x e. sn b <-> F @ x = b
8 7 imeq2i 
x e. a -> F @ x e. sn b <-> x e. a -> F @ x = b
9 8 aleqi 
A. x (x e. a -> F @ x e. sn b) <-> A. x (x e. a -> F @ x = b)
10 9 aneq2i 
isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b) <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x = b)
11 6, 10 ax_mp 
F == \. x e. a, b <-> isfun F /\ Dom F == a /\ A. x (x e. a -> F @ x e. sn b)
12 3, 11 ax_mp 
func F a (sn b) <-> F == \. x e. a, b

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)