Theorem funcappb ≪ | index | src | ≫ 

theorem funcappb (A B F: set) (a b: nat):
  $ func F A B -> (a, b e. F <-> a e. A /\ F @ a = b) $;
StepHypRefExpression
1 isfappb 
isfun F -> (a, b e. F <-> a e. Dom F /\ F @ a = b)
2 funcisf 
func F A B -> isfun F
3 1, 2 syl 
func F A B -> (a, b e. F <-> a e. Dom F /\ F @ a = b)
4 funcdm 
func F A B -> Dom F == A
5 4 eleq2d 
func F A B -> (a e. Dom F <-> a e. A)
6 5 aneq1d 
func F A B -> (a e. Dom F /\ F @ a = b <-> a e. A /\ F @ a = b)
7 3, 6 bitrd 
func F A B -> (a, b e. F <-> a e. A /\ F @ 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 (peano2, addeq, muleq)