Theorem resapp ≪ | index | src | ≫

theorem resapp (A F: set) (a: nat): $ a e. A -> (F |` A) @ a = F @ a $;

Step	Hyp	Ref	Expression
1		eqapp	A. y (a, y e. F \|` A <-> a, y e. F) -> (F \|` A) @ a = F @ a
2		prelres	a, y e. F \|` A <-> a, y e. F /\ a e. A
3		bian2	a e. A -> (a, y e. F /\ a e. A <-> a, y e. F)
4	2, 3	syl5bb	a e. A -> (a, y e. F \|` A <-> a, y e. F)
5	4	iald	a e. A -> A. y (a, y e. F \|` A <-> a, y e. F)
6	1, 5	syl	a e. A -> (F \|` A) @ a = 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)