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Theorem breldmd 5751
 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 5748 . 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 2111   class class class wbr 5034  dom cdm 5523 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-un 3888  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-dm 5533 This theorem is referenced by:  fvelimad  6717  climresdm  42660  xlimliminflimsup  42672
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