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Theorem breldmg 5901
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 5145 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21spcegv 3584 . . . 4 (𝐵𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
32imp 407 . . 3 ((𝐵𝐷𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
4 eldmg 5890 . . 3 (𝐴𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
53, 4imbitrrid 245 . 2 (𝐴𝐶 → ((𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅))
653impib 1116 1 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wex 1781  wcel 2106   class class class wbr 5141  dom cdm 5669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-dm 5679
This theorem is referenced by:  breldmd  5904  brelrng  5932  releldm  5935  sossfld  6174  brtpos  8202  fprresex  8277  wfrlem17OLD  8307  tfrlem9a  8368  perpln1  27826  lmdvg  32764  esumcvgsum  32917  climeldmeq  44154  climfv  44180  climxlim2  44335  sge0isum  44916  smflimsuplem6  45314  eubrdm  45518  funressneu  45529  tz6.12-afv  45653  rlimdmafv  45657  tz6.12-afv2  45720  rlimdmafv2  45738
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