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Theorem breldmd 5869
Description: Membership of first of a binary relation in a domain. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
Hypotheses
Ref Expression
breldmd.1 (𝜑𝐴𝐶)
breldmd.2 (𝜑𝐵𝐷)
breldmd.3 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
breldmd (𝜑𝐴 ∈ dom 𝑅)

Proof of Theorem breldmd
StepHypRef Expression
1 breldmd.1 . 2 (𝜑𝐴𝐶)
2 breldmd.2 . 2 (𝜑𝐵𝐷)
3 breldmd.3 . 2 (𝜑𝐴𝑅𝐵)
4 breldmg 5866 . 2 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
51, 2, 3, 4syl3anc 1372 1 (𝜑𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5106  dom cdm 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-dm 5644
This theorem is referenced by:  fvelimad  6910  climresdm  44177  xlimliminflimsup  44189
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