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Theorem opeldmd 5249
 Description: Membership of first of an ordered pair in a domain. Deduction version of opeldm 5250. (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 4341 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2672 . . . 4 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
43spcegv 3267 . . 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 5242 . . 3 (𝐴𝑉 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
86, 7syl 17 . 2 (𝜑 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
95, 8sylibrd 248 1 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131  dom cdm 5038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-dm 5048 This theorem is referenced by:  eupth2eucrct  41385
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