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Theorem releldmb 4863
Description: Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
Assertion
Ref Expression
releldmb (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem releldmb
StepHypRef Expression
1 eldmg 4821 . . 3 (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
21ibi 176 . 2 (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3 releldm 4861 . . . 4 ((Rel 𝑅𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
43ex 115 . . 3 (Rel 𝑅 → (𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
54exlimdv 1819 . 2 (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
62, 5impbid2 143 1 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wex 1492  wcel 2148   class class class wbr 4002  dom cdm 4625  Rel wrel 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-dm 4635
This theorem is referenced by: (None)
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