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Theorem relelfvdm 5097
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 (𝐹B)) → B dom 𝐹)

Proof of Theorem relelfvdm
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfv 5068 . . . . . 6 (A (𝐹B) ↔ x(A x y(B𝐹yy = x)))
2 exsimpr 1492 . . . . . 6 (x(A x y(B𝐹yy = x)) → xy(B𝐹yy = x))
31, 2sylbi 114 . . . . 5 (A (𝐹B) → xy(B𝐹yy = x))
4 equsb1 1650 . . . . . . . 8 [x / y]y = x
5 spsbbi 1707 . . . . . . . 8 (y(B𝐹yy = x) → ([x / y]B𝐹y ↔ [x / y]y = x))
64, 5mpbiri 157 . . . . . . 7 (y(B𝐹yy = x) → [x / y]B𝐹y)
7 nfv 1403 . . . . . . . 8 y B𝐹x
8 breq2 3720 . . . . . . . 8 (y = x → (B𝐹yB𝐹x))
97, 8sbie 1656 . . . . . . 7 ([x / y]B𝐹yB𝐹x)
106, 9sylib 127 . . . . . 6 (y(B𝐹yy = x) → B𝐹x)
1110eximi 1474 . . . . 5 (xy(B𝐹yy = x) → x B𝐹x)
123, 11syl 14 . . . 4 (A (𝐹B) → x B𝐹x)
1312anim2i 324 . . 3 ((Rel 𝐹 A (𝐹B)) → (Rel 𝐹 x B𝐹x))
14 19.42v 1769 . . 3 (x(Rel 𝐹 B𝐹x) ↔ (Rel 𝐹 x B𝐹x))
1513, 14sylibr 137 . 2 ((Rel 𝐹 A (𝐹B)) → x(Rel 𝐹 B𝐹x))
16 releldm 4462 . . 3 ((Rel 𝐹 B𝐹x) → B dom 𝐹)
1716exlimiv 1472 . 2 (x(Rel 𝐹 B𝐹x) → B dom 𝐹)
1815, 17syl 14 1 ((Rel 𝐹 A (𝐹B)) → B dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1314  wex 1361   wcel 1375  [wsb 1627   class class class wbr 3716  dom cdm 4238  Rel wrel 4243  cfv 4796
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-xp 4244  df-rel 4245  df-dm 4248  df-iota 4761  df-fv 4804
This theorem is referenced by:  elmpt2cl  5587  mpt2xopn0yelv  5742
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