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Theorem relfvssunirn 5550
Description: The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
relfvssunirn  |-  ( Rel 
F  ->  ( F `  A )  C_  U. ran  F )

Proof of Theorem relfvssunirn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 relelrn 4881 . . . . 5  |-  ( ( Rel  F  /\  A F x )  ->  x  e.  ran  F )
21ex 115 . . . 4  |-  ( Rel 
F  ->  ( A F x  ->  x  e. 
ran  F ) )
3 elssuni 3852 . . . 4  |-  ( x  e.  ran  F  ->  x  C_  U. ran  F
)
42, 3syl6 33 . . 3  |-  ( Rel 
F  ->  ( A F x  ->  x  C_  U.
ran  F ) )
54alrimiv 1885 . 2  |-  ( Rel 
F  ->  A. x
( A F x  ->  x  C_  U. ran  F ) )
6 fvss 5548 . 2  |-  ( A. x ( A F x  ->  x  C_  U. ran  F )  ->  ( F `  A )  C_  U. ran  F )
75, 6syl 14 1  |-  ( Rel 
F  ->  ( F `  A )  C_  U. ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1362    e. wcel 2160    C_ wss 3144   U.cuni 3824   class class class wbr 4018   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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
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:  relrnfvex  5552
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