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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 5769. (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 4796 | . . . . 5 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
3 | 2 | eleq1d 2894 | . . . 4 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
4 | 3 | spcegv 3594 | . . 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 5761 | . . 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 260 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∃wex 1771 ∈ wcel 2105 〈cop 4563 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-dm 5558 |
This theorem is referenced by: eupth2eucrct 27923 |
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