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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 5778. (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 4806 | . . . . 5 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
3 | 2 | eleq1d 2899 | . . . 4 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
4 | 3 | spcegv 3599 | . . 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 5770 | . . 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 261 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 〈cop 4575 dom cdm 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-dm 5567 |
This theorem is referenced by: eupth2eucrct 27998 |
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