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Theorem afvvfveq 27161
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
afvvfveq  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )

Proof of Theorem afvvfveq
StepHypRef Expression
1 nvel 4190 . . . 4  |-  -.  _V  e.  B
2 eleq1 2376 . . . . 5  |-  ( _V  =  ( F''' A )  ->  ( _V  e.  B 
<->  ( F''' A )  e.  B
) )
32eqcoms 2319 . . . 4  |-  ( ( F''' A )  =  _V  ->  ( _V  e.  B  <->  ( F''' A )  e.  B
) )
41, 3mtbii 293 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
54necon2ai 2524 . 2  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =/=  _V )
6 afvnufveq 27160 . 2  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
75, 6syl 15 1  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701    =/= wne 2479   _Vcvv 2822   ` cfv 5292  '''cafv 27120
This theorem is referenced by:  afv0fv0  27162  afv0nbfvbi  27164  aovvoveq  27205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rab 2586  df-v 2824  df-un 3191  df-if 3600  df-fv 5300  df-afv 27123
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