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Theorem afvfvn0fveq 27971
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 5754 . . 3  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
2 df-dfat 27931 . . 3  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
31, 2sylibr 204 . 2  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
4 afvfundmfveq 27959 . 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 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   {csn 3806   dom cdm 4870    |` cres 4872   Fun wfun 5440   ` cfv 5446   defAt wdfat 27928  '''cafv 27929
This theorem is referenced by:  aovovn0oveq  28015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-dfat 27931  df-afv 27932
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