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Theorem releldmb 4776
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 4734 . . 3 (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
21ibi 175 . 2 (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3 releldm 4774 . . . 4 ((Rel 𝑅𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
43ex 114 . . 3 (Rel 𝑅 → (𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
54exlimdv 1791 . 2 (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥𝐴 ∈ dom 𝑅))
62, 5impbid2 142 1 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wex 1468  wcel 1480   class class class wbr 3929  dom cdm 4539  Rel wrel 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-dm 4549
This theorem is referenced by: (None)
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