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Mirrors > Home > ILE Home > Th. List > opeldm | 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 3550 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
3 | 2 | eleq1d 2106 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
4 | 1, 3 | spcev 2647 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶) |
5 | opeldm.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | eldm2 4533 | . 2 ⊢ (𝐴 ∈ dom 𝐶 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶) |
7 | 4, 6 | sylibr 137 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 〈cop 3378 dom cdm 4345 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-dm 4355 |
This theorem is referenced by: breldm 4539 elreldm 4560 relssres 4648 iss 4654 imadmrn 4678 dfco2a 4821 funssres 4942 funun 4944 iinerm 6178 |
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