Theorem rlamapp ≪ | index | src | ≫ 

theorem rlamapp (F: set) (a: nat) {x: nat}:
  $ \. x e. a, F @ x == F <-> isfun F /\ Dom F == a $;
StepHypRefExpression
1 bitr 
(\. x e. a, F @ x == F <-> (\ x, F @ x) |` a == F) -> ((\ x, F @ x) |` a == F <-> isfun F /\ Dom F == a) -> (\. x e. a, F @ x == F <-> isfun F /\ Dom F == a)
2 eqseq1 
\. x e. a, F @ x == (\ x, F @ x) |` a -> (\. x e. a, F @ x == F <-> (\ x, F @ x) |` a == F)
3 rlameqs 
\. x e. a, F @ x == (\ x, F @ x) |` a
4 2, 3 ax_mp 
\. x e. a, F @ x == F <-> (\ x, F @ x) |` a == F
5 1, 4 ax_mp 
((\ x, F @ x) |` a == F <-> isfun F /\ Dom F == a) -> (\. x e. a, F @ x == F <-> isfun F /\ Dom F == a)
6 lamapp2 
(\ x, F @ x) |` a == F <-> isfun F /\ Dom F == a
7 5, 6 ax_mp 
\. x e. a, F @ x == F <-> isfun F /\ Dom F == a

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)