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Theorem opeldmd 5768
Description: Membership of first of an ordered pair in a domain. Deduction version of opeldm 5769. (Contributed by AV, 11-Mar-2021.)
Hypotheses
Ref Expression
opeldmd.1 (𝜑𝐴𝑉)
opeldmd.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
opeldmd (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))

Proof of Theorem opeldmd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 opeldmd.2 . . 3 (𝜑𝐵𝑊)
2 opeq2 4796 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2894 . . . 4 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
43spcegv 3594 . . 3 (𝐵𝑊 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
51, 4syl 17 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
6 opeldmd.1 . . 3 (𝜑𝐴𝑉)
7 eldm2g 5761 . . 3 (𝐴𝑉 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
86, 7syl 17 . 2 (𝜑 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
95, 8sylibrd 260 1 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wex 1771  wcel 2105  cop 4563  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:  eupth2eucrct  27923
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