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Theorem breldmg 5923
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 5152 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21spcegv 3597 . . . 4 (𝐵𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
32imp 406 . . 3 ((𝐵𝐷𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
4 eldmg 5912 . . 3 (𝐴𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
53, 4imbitrrid 246 . 2 (𝐴𝐶 → ((𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅))
653impib 1115 1 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wex 1776  wcel 2106   class class class wbr 5148  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699
This theorem is referenced by:  breldmd  5926  brelrng  5955  releldm  5958  sossfld  6208  brtpos  8259  fprresex  8334  wfrlem17OLD  8364  tfrlem9a  8425  perpln1  28733  lmdvg  33914  esumcvgsum  34069  climeldmeq  45621  climfv  45647  climxlim2  45802  sge0isum  46383  smflimsuplem6  46781  eubrdm  46986  funressneu  46997  tz6.12-afv  47123  rlimdmafv  47127  tz6.12-afv2  47190  rlimdmafv2  47208
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