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Theorem breldmg 4745
 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 3933 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21spcegv 2774 . . . 4 (𝐵𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
32imp 123 . . 3 ((𝐵𝐷𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
433adant1 999 . 2 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
5 eldmg 4734 . . 3 (𝐴𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
653ad2ant1 1002 . 2 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
74, 6mpbird 166 1 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 962  ∃wex 1468   ∈ wcel 1480   class class class wbr 3929  dom cdm 4539 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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 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-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549 This theorem is referenced by:  brelrng  4770  releldm  4774  brtposg  6151  shftfvalg  10602  shftfval  10605  geolim2  11293  geoisum1c  11301  ntrivcvgap  11329  eftlub  11408  eflegeo  11419  dvcj  12856  dvrecap  12860  dvef  12871  trilpolemisumle  13292  trilpolemeq1  13294
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