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Theorem relfvssunirn 5389
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 4733 . . . . 5  |-  ( ( Rel  F  /\  A F x )  ->  x  e.  ran  F )
21ex 114 . . . 4  |-  ( Rel 
F  ->  ( A F x  ->  x  e. 
ran  F ) )
3 elssuni 3728 . . . 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 1826 . 2  |-  ( Rel 
F  ->  A. x
( A F x  ->  x  C_  U. ran  F ) )
6 fvss 5387 . 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 1310    e. wcel 1461    C_ wss 3035   U.cuni 3700   class class class wbr 3893   ran crn 4498   Rel wrel 4502   ` cfv 5079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-cnv 4505  df-dm 4507  df-rn 4508  df-iota 5044  df-fv 5087
This theorem is referenced by:  relrnfvex  5391
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