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Theorem afvvfveq 28011
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 4153 . . . 4  |-  -.  _V  e.  B
2 eleq1 2343 . . . . 5  |-  ( _V  =  ( F''' A )  ->  ( _V  e.  B 
<->  ( F''' A )  e.  B
) )
32eqcoms 2286 . . . 4  |-  ( ( F''' A )  =  _V  ->  ( _V  e.  B  <->  ( F''' A )  e.  B
) )
41, 3mtbii 293 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
54necon2ai 2491 . 2  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =/=  _V )
6 afvnufveq 28010 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   ` cfv 5255  '''cafv 27972
This theorem is referenced by:  afv0fv0  28012  afv0nbfvbi  28014  aovvoveq  28052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-afv 27975
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