Theorem funcT ≪ | index | src | ≫ 

theorem funcT (A B F: set) (x: nat): $ func F A B /\ x e. A -> F @ x e. B $;
StepHypRefExpression
1 funcrn 
func F A B -> Ran F C_ B
2 1 anwl 
func F A B /\ x e. A -> Ran F C_ B
3 appelrn 
isfun F -> x e. Dom F -> F @ x e. Ran F
4 funcisf 
func F A B -> isfun F
5 4 anwl 
func F A B /\ x e. A -> isfun F
6 funcdm 
func F A B -> Dom F == A
7 6 eleq2d 
func F A B -> (x e. Dom F <-> x e. A)
8 7 bi2d 
func F A B -> x e. A -> x e. Dom F
9 8 imp 
func F A B /\ x e. A -> x e. Dom F
10 3, 5, 9 sylc 
func F A B /\ x e. A -> F @ x e. Ran F
11 2, 10 sseld 
func F A B /\ x e. A -> F @ x e. 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)