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Theorem relelfvdm 5320
Description: If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)
Assertion
Ref Expression
relelfvdm  |-  ( ( Rel  F  /\  A  e.  ( F `  B
) )  ->  B  e.  dom  F )

Proof of Theorem relelfvdm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfv 5287 . . . . . 6  |-  ( A  e.  ( F `  B )  <->  E. x
( A  e.  x  /\  A. y ( B F y  <->  y  =  x ) ) )
2 exsimpr 1554 . . . . . 6  |-  ( E. x ( A  e.  x  /\  A. y
( B F y  <-> 
y  =  x ) )  ->  E. x A. y ( B F y  <->  y  =  x ) )
31, 2sylbi 119 . . . . 5  |-  ( A  e.  ( F `  B )  ->  E. x A. y ( B F y  <->  y  =  x ) )
4 equsb1 1715 . . . . . . . 8  |-  [ x  /  y ] y  =  x
5 spsbbi 1772 . . . . . . . 8  |-  ( A. y ( B F y  <->  y  =  x )  ->  ( [
x  /  y ] B F y  <->  [ x  /  y ] y  =  x ) )
64, 5mpbiri 166 . . . . . . 7  |-  ( A. y ( B F y  <->  y  =  x )  ->  [ x  /  y ] B F y )
7 nfv 1466 . . . . . . . 8  |-  F/ y  B F x
8 breq2 3841 . . . . . . . 8  |-  ( y  =  x  ->  ( B F y  <->  B F x ) )
97, 8sbie 1721 . . . . . . 7  |-  ( [ x  /  y ] B F y  <->  B F x )
106, 9sylib 120 . . . . . 6  |-  ( A. y ( B F y  <->  y  =  x )  ->  B F x )
1110eximi 1536 . . . . 5  |-  ( E. x A. y ( B F y  <->  y  =  x )  ->  E. x  B F x )
123, 11syl 14 . . . 4  |-  ( A  e.  ( F `  B )  ->  E. x  B F x )
1312anim2i 334 . . 3  |-  ( ( Rel  F  /\  A  e.  ( F `  B
) )  ->  ( Rel  F  /\  E. x  B F x ) )
14 19.42v 1834 . . 3  |-  ( E. x ( Rel  F  /\  B F x )  <-> 
( Rel  F  /\  E. x  B F x ) )
1513, 14sylibr 132 . 2  |-  ( ( Rel  F  /\  A  e.  ( F `  B
) )  ->  E. x
( Rel  F  /\  B F x ) )
16 releldm 4658 . . 3  |-  ( ( Rel  F  /\  B F x )  ->  B  e.  dom  F )
1716exlimiv 1534 . 2  |-  ( E. x ( Rel  F  /\  B F x )  ->  B  e.  dom  F )
1815, 17syl 14 1  |-  ( ( Rel  F  /\  A  e.  ( F `  B
) )  ->  B  e.  dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426    e. wcel 1438   [wsb 1692   class class class wbr 3837   dom cdm 4428   Rel wrel 4433   ` cfv 5002
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-dm 4438  df-iota 4967  df-fv 5010
This theorem is referenced by:  elmpt2cl  5824  mpt2xopn0yelv  5986  eluzel2  8993  hashinfom  10151
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