Theorem eqres ≪ | index | src | ≫

theorem eqres (A F: set): $ Dom F C_ A <-> F |` A == F $;

Step	Hyp	Ref	Expression
1		bitr	(Dom F C_ A <-> F C_ Xp A _V) -> (F C_ Xp A _V <-> F \|` A == F) -> (Dom F C_ A <-> F \|` A == F)
2		ssdm	Dom F C_ A <-> F C_ Xp A _V
3	1, 2	ax_mp	(F C_ Xp A _V <-> F \|` A == F) -> (Dom F C_ A <-> F \|` A == F)
4		eqin1	F C_ Xp A _V <-> F i^i Xp A _V == F
5	4	conv res	F C_ Xp A _V <-> F \|` A == F
6	3, 5	ax_mp	Dom F C_ A <-> F \|` A == 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)