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Theorem fvssunirng 5390
Description: The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
fvssunirng  |-  ( A  e.  _V  ->  ( F `  A )  C_ 
U. ran  F )

Proof of Theorem fvssunirng
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2660 . . . . 5  |-  x  e. 
_V
2 brelrng 4730 . . . . . 6  |-  ( ( A  e.  _V  /\  x  e.  _V  /\  A F x )  ->  x  e.  ran  F )
323exp 1163 . . . . 5  |-  ( A  e.  _V  ->  (
x  e.  _V  ->  ( A F x  ->  x  e.  ran  F ) ) )
41, 3mpi 15 . . . 4  |-  ( A  e.  _V  ->  ( A F x  ->  x  e.  ran  F ) )
5 elssuni 3730 . . . 4  |-  ( x  e.  ran  F  ->  x  C_  U. ran  F
)
64, 5syl6 33 . . 3  |-  ( A  e.  _V  ->  ( A F x  ->  x  C_ 
U. ran  F )
)
76alrimiv 1828 . 2  |-  ( A  e.  _V  ->  A. x
( A F x  ->  x  C_  U. ran  F ) )
8 fvss 5389 . 2  |-  ( A. x ( A F x  ->  x  C_  U. ran  F )  ->  ( F `  A )  C_  U. ran  F )
97, 8syl 14 1  |-  ( A  e.  _V  ->  ( F `  A )  C_ 
U. ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    e. wcel 1463   _Vcvv 2657    C_ wss 3037   U.cuni 3702   class class class wbr 3895   ran crn 4500   ` cfv 5081
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-cnv 4507  df-dm 4509  df-rn 4510  df-iota 5046  df-fv 5089
This theorem is referenced by:  fvexg  5394  strfvssn  11824  xmetunirn  12347  mopnval  12431
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