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Mirrors > Home > ILE Home > Th. List > opeldmg | GIF version |
Description: Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.) |
Ref | Expression |
---|---|
opeldmg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 3714 | . . . . 5 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
2 | 1 | eleq1d 2209 | . . . 4 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
3 | 2 | spcegv 2777 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (〈𝐴, 𝐵〉 ∈ 𝐶 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶)) |
4 | 3 | adantl 275 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ 𝐶 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶)) |
5 | eldm2g 4743 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶)) | |
6 | 5 | adantr 274 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐶)) |
7 | 4, 6 | sylibrd 168 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∃wex 1469 ∈ wcel 1481 〈cop 3535 dom cdm 4547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-dm 4557 |
This theorem is referenced by: tfr0dm 6227 tfrlemi14d 6238 tfr1onlemres 6254 tfrcllemres 6267 fnfi 6833 frecuzrdgtcl 10216 frecuzrdgdomlem 10221 hashennn 10558 |
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