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Theorem relrnfvex 5552
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  e.  _V )  -> 
( F `  A
)  e.  _V )

Proof of Theorem relrnfvex
StepHypRef Expression
1 relfvssunirn 5550 . 2  |-  ( Rel 
F  ->  ( F `  A )  C_  U. ran  F )
2 uniexg 4457 . 2  |-  ( ran 
F  e.  _V  ->  U.
ran  F  e.  _V )
3 ssexg 4157 . 2  |-  ( ( ( F `  A
)  C_  U. ran  F  /\  U. ran  F  e. 
_V )  ->  ( F `  A )  e.  _V )
41, 2, 3syl2an 289 1  |-  ( ( Rel  F  /\  ran  F  e.  _V )  -> 
( F `  A
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   _Vcvv 2752    C_ wss 3144   U.cuni 3824   ran crn 4645   Rel wrel 4649   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-iota 5196  df-fv 5243
This theorem is referenced by: (None)
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