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Theorem afvfvn0fveq 28118
Description: If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvfvn0fveq  |-  ( ( F `  A )  =/=  (/)  ->  ( F''' A )  =  ( F `
 A ) )

Proof of Theorem afvfvn0fveq
StepHypRef Expression
1 fvfundmfvn0 28091 . . 3  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
2 df-dfat 28077 . . 3  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
31, 2sylibr 203 . 2  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
4 afvfundmfveq 28106 . 2  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
53, 4syl 15 1  |-  ( ( F `  A )  =/=  (/)  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   {csn 3653   dom cdm 4705    |` cres 4707   Fun wfun 5265   ` cfv 5271   defAt wdfat 28074  '''cafv 28075
This theorem is referenced by:  aovovn0oveq  28162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-dfat 28077  df-afv 28078
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