Theorem opeldmd 5327

Description: Membership of first of an ordered pair in a domain. Deduction version of opeldm 5328. (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 4403	. . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3	2	eleq1d 2686	. . . 4 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	3	spcegv 3294	. . 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 5320	. . 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 249	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ⟨cop 4183  dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-dm 5124

This theorem is referenced by:  eupth2eucrct  27077