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Theorem afvnfundmuv 28013
Description: If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
afvnfundmuv  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )

Proof of Theorem afvnfundmuv
StepHypRef Expression
1 dfafv2 28006 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iffalse 3574 . 2  |-  ( -.  F defAt  A  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  _V )
31, 2syl5eq 2329 1  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1625   _Vcvv 2790   ifcif 3567   ` cfv 5257   defAt wdfat 27982  '''cafv 27983
This theorem is referenced by:  ndmafv  28014  nfunsnafv  28016  afvnufveq  28021  afvres  28045  afvco2  28048  aovnfundmuv  28053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-xp 4697  df-cnv 4699  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fv 5265  df-afv 27986
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