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Theorem breldmg 4568
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
Assertion
Ref Expression
breldmg ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Proof of Theorem breldmg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq2 3795 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21spcegv 2658 . . . 4 (𝐵𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
32imp 119 . . 3 ((𝐵𝐷𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
433adant1 933 . 2 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
5 eldmg 4557 . . 3 (𝐴𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
653ad2ant1 936 . 2 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
74, 6mpbird 160 1 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  w3a 896  wex 1397  wcel 1409   class class class wbr 3791  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-dm 4382
This theorem is referenced by:  brelrng  4592  releldm  4596  brtposg  5899  shftfvalg  9646  shftfval  9649
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