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Theorem relelfvdm 5237
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 5207 . . . . . 6  |-  ( A  e.  ( F `  B )  <->  E. x
( A  e.  x  /\  A. y ( B F y  <->  y  =  x ) ) )
2 exsimpr 1550 . . . . . 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 1709 . . . . . . . 8  |-  [ x  /  y ] y  =  x
5 spsbbi 1766 . . . . . . . 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 1462 . . . . . . . 8  |-  F/ y  B F x
8 breq2 3797 . . . . . . . 8  |-  ( y  =  x  ->  ( B F y  <->  B F x ) )
97, 8sbie 1715 . . . . . . 7  |-  ( [ x  /  y ] B F y  <->  B F x )
106, 9sylib 120 . . . . . 6  |-  ( A. y ( B F y  <->  y  =  x )  ->  B F x )
1110eximi 1532 . . . . 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 1828 . . 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 4597 . . 3  |-  ( ( Rel  F  /\  B F x )  ->  B  e.  dom  F )
1716exlimiv 1530 . 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 1283   E.wex 1422    e. wcel 1434   [wsb 1686   class class class wbr 3793   dom cdm 4371   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-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-dm 4381  df-iota 4897  df-fv 4940
This theorem is referenced by:  elmpt2cl  5729  mpt2xopn0yelv  5888  eluzel2  8705  sizeinf  9802
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