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Theorem relfvssunirn 5222
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 4598 . . . . 5  |-  ( ( Rel  F  /\  A F x )  ->  x  e.  ran  F )
21ex 113 . . . 4  |-  ( Rel 
F  ->  ( A F x  ->  x  e. 
ran  F ) )
3 elssuni 3637 . . . 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 1796 . 2  |-  ( Rel 
F  ->  A. x
( A F x  ->  x  C_  U. ran  F ) )
6 fvss 5220 . 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 1283    e. wcel 1434    C_ wss 2974   U.cuni 3609   class class class wbr 3793   ran crn 4372   Rel wrel 4376   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-iota 4897  df-fv 4940
This theorem is referenced by:  relrnfvex  5224
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