Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funALTVeq Structured version   Visualization version   GIF version

Theorem funALTVeq 38302
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
StepHypRef Expression
1 eqimss2 4036 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 funALTVss 38301 . . 3 (𝐵𝐴 → ( FunALTV 𝐴 → FunALTV 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → ( FunALTV 𝐴 → FunALTV 𝐵))
4 eqimss 4035 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 funALTVss 38301 . . 3 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
73, 6impbid 211 1 (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wss 3944   FunALTV wfunALTV 37810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-coss 38013  df-cnvrefrel 38129  df-funALTV 38284
This theorem is referenced by:  funALTVeqi  38303  funALTVeqd  38304
  Copyright terms: Public domain W3C validator