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Theorem relelfvdm 5148
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 5119 . . . . . 6  F `  F
2 exsimpr 1506 . . . . . 6  F  F
31, 2sylbi 114 . . . . 5  F `  F
4 equsb1 1665 . . . . . . . 8
5 spsbbi 1722 . . . . . . . 8  F  F
64, 5mpbiri 157 . . . . . . 7  F  F
7 nfv 1418 . . . . . . . 8  F/  F
8 breq2 3759 . . . . . . . 8  F  F
97, 8sbie 1671 . . . . . . 7  F  F
106, 9sylib 127 . . . . . 6  F  F
1110eximi 1488 . . . . 5  F  F
123, 11syl 14 . . . 4  F `  F
1312anim2i 324 . . 3  Rel  F  F `  Rel  F  F
14 19.42v 1783 . . 3  Rel  F  F  Rel 
F  F
1513, 14sylibr 137 . 2  Rel  F  F `  Rel  F  F
16 releldm 4512 . . 3  Rel  F  F  dom  F
1716exlimiv 1486 . 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 1240  wex 1378   wcel 1390  wsb 1642   class class class wbr 3755   dom cdm 4288   Rel wrel 4293   ` cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-iota 4810  df-fv 4853
This theorem is referenced by:  elmpt2cl  5640  mpt2xopn0yelv  5795  eluzel2  8214
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