Theorem funALTVeq 36373

Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
funALTVeq	⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

Proof of Theorem funALTVeq

Step	Hyp	Ref	Expression
1		eqimss2 3949	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funALTVss 36372	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ( FunALTV 𝐴 → FunALTV 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 → FunALTV 𝐵))
4		eqimss 3948	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funALTVss 36372	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
7	3, 6	impbid 215	1 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ⊆ wss 3858   FunALTV wfunALTV 35924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-coss 36099  df-cnvrefrel 36205  df-funALTV 36355

This theorem is referenced by:  funALTVeqi  36374  funALTVeqd  36375