![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > opeldmd | Structured version Visualization version GIF version |
Description: Membership of first of an ordered pair in a domain. Deduction version of opeldm 5864. (Contributed by AV, 11-Mar-2021.) |
Ref | Expression |
---|---|
opeldmd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
opeldmd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
opeldmd | ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeldmd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | opeq2 4832 | . . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩) | |
3 | 2 | eleq1d 2819 | . . . 4 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)) |
4 | 3 | spcegv 3555 | . . 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 5856 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ⟨cop 4593 dom cdm 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-dm 5644 |
This theorem is referenced by: eupth2eucrct 29203 |
Copyright terms: Public domain | W3C validator |