Theorem lamapp ≪ | index | src | ≫

theorem lamapp (F: set) {x: nat}: $ (\ x, F @ x) |` Dom F == F <-> isfun F $;

Step	Hyp	Ref	Expression
1		bitr	((\ x, F @ x) \|` Dom F == F <-> isfun F /\ Dom F == Dom F) -> (isfun F /\ Dom F == Dom F <-> isfun F) -> ((\ x, F @ x) \|` Dom F == F <-> isfun F)
2		lamapp2	(\ x, F @ x) \|` Dom F == F <-> isfun F /\ Dom F == Dom F
3	1, 2	ax_mp	(isfun F /\ Dom F == Dom F <-> isfun F) -> ((\ x, F @ x) \|` Dom F == F <-> isfun F)
4		bian2	Dom F == Dom F -> (isfun F /\ Dom F == Dom F <-> isfun F)
5		eqsid	Dom F == Dom F
6	4, 5	ax_mp	isfun F /\ Dom F == Dom F <-> isfun F
7	3, 6	ax_mp	(\ x, F @ x) \|` Dom F == F <-> isfun F

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)