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Theorem breldmd 5774
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 5771 . 2 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
51, 2, 3, 4syl3anc 1363 1 (𝜑𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105   class class class wbr 5057  dom cdm 5548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-dm 5558
This theorem is referenced by:  fvelimad  6725  climresdm  42007  xlimliminflimsup  42019
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