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Theorem afvfvn0fveq 27685
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 27658 . . 3  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
2 df-dfat 27644 . . 3  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
31, 2sylibr 204 . 2  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
4 afvfundmfveq 27673 . 2  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
53, 4syl 16 1  |-  ( ( F `  A )  =/=  (/)  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   (/)c0 3573   {csn 3759   dom cdm 4820    |` cres 4822   Fun wfun 5390   ` cfv 5396   defAt wdfat 27641  '''cafv 27642
This theorem is referenced by:  aovovn0oveq  27729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-res 4832  df-iota 5360  df-fun 5398  df-fv 5404  df-dfat 27644  df-afv 27645
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