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Theorem relfvssunirn 5512
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 4847 . . . . 5  |-  ( ( Rel  F  /\  A F x )  ->  x  e.  ran  F )
21ex 114 . . . 4  |-  ( Rel 
F  ->  ( A F x  ->  x  e. 
ran  F ) )
3 elssuni 3824 . . . 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 1867 . 2  |-  ( Rel 
F  ->  A. x
( A F x  ->  x  C_  U. ran  F ) )
6 fvss 5510 . 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 1346    e. wcel 2141    C_ wss 3121   U.cuni 3796   class class class wbr 3989   ran crn 4612   Rel wrel 4616   ` cfv 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-iota 5160  df-fv 5206
This theorem is referenced by:  relrnfvex  5514
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