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Theorem breldmg 5252
 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 4587 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21spcegv 3267 . . . 4 (𝐵𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
32imp 444 . . 3 ((𝐵𝐷𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
4 eldmg 5241 . . 3 (𝐴𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
53, 4syl5ibr 235 . 2 (𝐴𝐶 → ((𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅))
653impib 1254 1 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583  dom cdm 5038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-dm 5048 This theorem is referenced by:  brelrng  5276  releldm  5279  sossfld  5499  brtpos  7248  wfrlem17  7318  tfrlem9a  7369  perpln1  25405  lmdvg  29327  esumcvgsum  29477  climeldmeq  38732  sge0isum  39320  tz6.12-afv  39902  rlimdmafv  39906
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