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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 4874 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
| 3 | 2 | eleq1d 2826 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
| 4 | 1, 3 | spcev 3606 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶) |
| 5 | opeldm.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 6 | 5 | eldm2 5912 | . 2 ⊢ (𝐴 ∈ dom 𝐶 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶) |
| 7 | 4, 6 | sylibr 234 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: breldm 5919 elreldm 5946 relssres 6040 iss 6053 imadmrn 6088 dfco2a 6266 relssdmrn 6288 funssres 6610 funun 6612 frxp2 8169 frxp3 8176 frrlem8 8318 frrlem10 8320 tz7.48-1 8483 iiner 8829 r0weon 10052 axdc3lem2 10491 uzrdgfni 13999 imasaddfnlem 17573 imasvscafn 17582 cicsym 17848 gsum2d 19990 noseqrdgfn 28312 cffldtocusgr 29464 cffldtocusgrOLD 29465 dfcnv2 32686 gsumfs2d 33058 bnj1379 34844 iss2 38345 rfovcnvf1od 44017 |
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