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Theorem afvnufveq 38405
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 )  =  ( F `  A ) )

Proof of Theorem afvnufveq
StepHypRef Expression
1 afvfundmfveq 38396 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
21con3i 141 . . 3  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  -.  F defAt  A )
3 afvnfundmuv 38397 . . 3  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
42, 3syl 17 . 2  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  ( F''' A )  =  _V )
54necon1ai 2656 1  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1438    =/= wne 2619   _Vcvv 3082   ` cfv 5599   defAt wdfat 38371  '''cafv 38372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-rab 2785  df-v 3084  df-un 3442  df-if 3911  df-fv 5607  df-afv 38375
This theorem is referenced by:  afvvfveq  38406  afvfv0bi  38410  aovnuoveq  38449
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