Theorem opeldmd 5920

Description: Membership of first of an ordered pair in a domain. Deduction version of opeldm 5921. (Contributed by AV, 11-Mar-2021.)

Hypotheses

Ref	Expression
opeldmd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
opeldmd.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
opeldmd	⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶))

Proof of Theorem opeldmd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeldmd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		opeq2 4879	. . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3	2	eleq1d 2824	. . . 4 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	3	spcegv 3597	. . 3 ⊢ (𝐵 ∈ 𝑊 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶))
5	1, 4	syl 17	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶))
6		opeldmd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		eldm2g 5913	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶))
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶))
9	5, 8	sylibrd 259	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ⟨cop 4637  dom cdm 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699

This theorem is referenced by:  eupth2eucrct  30246  tfsconcatb0  43334