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Theorem relelfvdm 5205
 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 𝐹𝐴 ∈ (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 5176 . . . . . 6 (𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
2 exsimpr 1509 . . . . . 6 (∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)) → ∃𝑥𝑦(𝐵𝐹𝑦𝑦 = 𝑥))
31, 2sylbi 114 . . . . 5 (𝐴 ∈ (𝐹𝐵) → ∃𝑥𝑦(𝐵𝐹𝑦𝑦 = 𝑥))
4 equsb1 1668 . . . . . . . 8 [𝑥 / 𝑦]𝑦 = 𝑥
5 spsbbi 1725 . . . . . . . 8 (∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥) → ([𝑥 / 𝑦]𝐵𝐹𝑦 ↔ [𝑥 / 𝑦]𝑦 = 𝑥))
64, 5mpbiri 157 . . . . . . 7 (∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥) → [𝑥 / 𝑦]𝐵𝐹𝑦)
7 nfv 1421 . . . . . . . 8 𝑦 𝐵𝐹𝑥
8 breq2 3768 . . . . . . . 8 (𝑦 = 𝑥 → (𝐵𝐹𝑦𝐵𝐹𝑥))
97, 8sbie 1674 . . . . . . 7 ([𝑥 / 𝑦]𝐵𝐹𝑦𝐵𝐹𝑥)
106, 9sylib 127 . . . . . 6 (∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥) → 𝐵𝐹𝑥)
1110eximi 1491 . . . . 5 (∃𝑥𝑦(𝐵𝐹𝑦𝑦 = 𝑥) → ∃𝑥 𝐵𝐹𝑥)
123, 11syl 14 . . . 4 (𝐴 ∈ (𝐹𝐵) → ∃𝑥 𝐵𝐹𝑥)
1312anim2i 324 . . 3 ((Rel 𝐹𝐴 ∈ (𝐹𝐵)) → (Rel 𝐹 ∧ ∃𝑥 𝐵𝐹𝑥))
14 19.42v 1786 . . 3 (∃𝑥(Rel 𝐹𝐵𝐹𝑥) ↔ (Rel 𝐹 ∧ ∃𝑥 𝐵𝐹𝑥))
1513, 14sylibr 137 . 2 ((Rel 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥(Rel 𝐹𝐵𝐹𝑥))
16 releldm 4569 . . 3 ((Rel 𝐹𝐵𝐹𝑥) → 𝐵 ∈ dom 𝐹)
1716exlimiv 1489 . 2 (∃𝑥(Rel 𝐹𝐵𝐹𝑥) → 𝐵 ∈ dom 𝐹)
1815, 17syl 14 1 ((Rel 𝐹𝐴 ∈ (𝐹𝐵)) → 𝐵 ∈ dom 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  [wsb 1645   class class class wbr 3764  dom cdm 4345  Rel wrel 4350  ‘cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355  df-iota 4867  df-fv 4910 This theorem is referenced by:  elmpt2cl  5698  mpt2xopn0yelv  5854  eluzel2  8478
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