Theorem funceq0 ≪ | index | src | ≫

theorem funceq0 (A B F: set): $ func F A B -> (F == 0 <-> A == 0) $;

Step	Hyp	Ref	Expression
1		dmeq0	Dom F == 0 <-> F == 0
2		funcdm	func F A B -> Dom F == A
3	2	eqseq1d	func F A B -> (Dom F == 0 <-> A == 0)
4	1, 3	syl5bbr	func F A B -> (F == 0 <-> A == 0)

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)