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Theorem relrnfvex 5118
Description: If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
relrnfvex  Rel  F  ran  F  _V  F `  _V

Proof of Theorem relrnfvex
StepHypRef Expression
1 relfvssunirn 5116 . 2  Rel 
F  F `  C_  U. ran  F
2 uniexg 4125 . 2  ran 
F  _V  U.
ran  F  _V
3 ssexg 3870 . 2  F `  C_  U. ran  F  U. ran  F 
_V  F `  _V
41, 2, 3syl2an 273 1  Rel  F  ran  F  _V  F `  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1374   _Vcvv 2535    C_ wss 2894   U.cuni 3554   ran crn 4273   Rel wrel 4277   ` cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-dm 4282  df-rn 4283  df-iota 4794  df-fv 4837
This theorem is referenced by: (None)
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