Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvnufveq Structured version   Visualization version   GIF version

Theorem afvnufveq 46662
Description: The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvnufveq ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))

Proof of Theorem afvnufveq
StepHypRef Expression
1 afvfundmfveq 46653 . . 3 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
2 afvnfundmuv 46654 . . 3 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
31, 2nsyl5 159 . 2 (¬ (𝐹'''𝐴) = (𝐹𝐴) → (𝐹'''𝐴) = V)
43necon1ai 2957 1 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wne 2929  Vcvv 3461  cfv 6549   defAt wdfat 46631  '''cafv 46632
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-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  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-uni 4910  df-int 4951  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-res 5690  df-iota 6501  df-fun 6551  df-fv 6557  df-aiota 46600  df-dfat 46634  df-afv 46635
This theorem is referenced by:  afvvfveq  46663  afvfv0bi  46667  aovnuoveq  46706
  Copyright terms: Public domain W3C validator