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Mirrors > Home > ILE Home > Th. List > optocl | GIF version |
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) |
Ref | Expression |
---|---|
optocl.1 | ⊢ 𝐷 = (𝐵 × 𝐶) |
optocl.2 | ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) |
optocl.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) |
Ref | Expression |
---|---|
optocl | ⊢ (𝐴 ∈ 𝐷 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp3 4593 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶))) | |
2 | opelxp 4569 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
3 | optocl.3 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) | |
4 | 2, 3 | sylbi 120 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) → 𝜑) |
5 | optocl.2 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | syl5ib 153 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) → 𝜓)) |
7 | 6 | imp 123 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) → 𝜓) |
8 | 7 | exlimivv 1868 | . . 3 ⊢ (∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) → 𝜓) |
9 | 1, 8 | sylbi 120 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝜓) |
10 | optocl.1 | . 2 ⊢ 𝐷 = (𝐵 × 𝐶) | |
11 | 9, 10 | eleq2s 2234 | 1 ⊢ (𝐴 ∈ 𝐷 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 〈cop 3530 × cxp 4537 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 |
This theorem is referenced by: 2optocl 4616 3optocl 4617 ecoptocl 6516 ax1rid 7685 ax0id 7686 axcnre 7689 |
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