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Theorem breldmd 5917
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 5914 . 2 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
51, 2, 3, 4syl3anc 1368 1 (𝜑𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5150  dom cdm 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-dm 5690
This theorem is referenced by:  fvelimad  6969  climresdm  45240  xlimliminflimsup  45252
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