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Theorem opeldmd 5233
Description: Membership of first of an ordered pair in a domain. Deduction version of opeldm 5234. (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 4332 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2668 . . . 4 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
43spcegv 3263 . . 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 5226 . . 3 (𝐴𝑉 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
86, 7syl 17 . 2 (𝜑 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
95, 8sylibrd 247 1 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wex 1694  wcel 1976  cop 4127  dom cdm 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-dm 5035
This theorem is referenced by:  eupth2eucrct  41384
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