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Theorem relelfvdm 5130
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  F `  dom  F

Proof of Theorem relelfvdm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfv 5101 . . . . . 6  F `  F
2 exsimpr 1491 . . . . . 6  F  F
31, 2sylbi 114 . . . . 5  F `  F
4 equsb1 1650 . . . . . . . 8
5 spsbbi 1707 . . . . . . . 8  F  F
64, 5mpbiri 157 . . . . . . 7  F  F
7 nfv 1402 . . . . . . . 8  F/  F
8 breq2 3742 . . . . . . . 8  F  F
97, 8sbie 1656 . . . . . . 7  F  F
106, 9sylib 127 . . . . . 6  F  F
1110eximi 1473 . . . . 5  F  F
123, 11syl 14 . . . 4  F `  F
1312anim2i 324 . . 3  Rel  F  F `  Rel  F  F
14 19.42v 1768 . . 3  Rel  F  F  Rel 
F  F
1513, 14sylibr 137 . 2  Rel  F  F `  Rel  F  F
16 releldm 4496 . . 3  Rel  F  F  dom  F
1716exlimiv 1471 . 2  Rel  F  F  dom  F
1815, 17syl 14 1  Rel  F  F `  dom  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1226  wex 1362   wcel 1374  wsb 1627   class class class wbr 3738   dom cdm 4272   Rel wrel 4277   ` cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-dm 4282  df-iota 4794  df-fv 4837
This theorem is referenced by:  elmpt2cl  5621  mpt2xopn0yelv  5776
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