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Mirrors > Home > MPE Home > Th. List > opeldm | Structured version Visualization version GIF version |
Description: Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.) |
Ref | Expression |
---|---|
opeldm.1 | ⊢ 𝐴 ∈ V |
opeldm.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opeldm | ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeldm.2 | . . 3 ⊢ 𝐵 ∈ V | |
2 | opeq2 4341 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
3 | 2 | eleq1d 2672 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
4 | 1, 3 | spcev 3273 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶) |
5 | opeldm.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | eldm2 5244 | . 2 ⊢ (𝐴 ∈ dom 𝐶 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶) |
7 | 4, 6 | sylibr 223 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 〈cop 4131 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 |
This theorem is referenced by: breldm 5251 elreldm 5271 relssres 5357 iss 5367 imadmrn 5395 dfco2a 5552 funssres 5844 funun 5846 tz7.48-1 7425 iiner 7706 r0weon 8718 axdc3lem2 9156 uzrdgfni 12619 imasaddfnlem 16011 imasvscafn 16020 cicsym 16287 gsum2d 18194 dfcnv2 28859 bnj1379 30155 rfovcnvf1od 37318 |
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