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Mirrors > Home > ILE Home > Th. List > oprab4 | GIF version |
Description: Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.) |
Ref | Expression |
---|---|
oprab4 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 4628 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | 1 | anbi1i 454 | . 2 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
3 | 2 | oprabbii 5888 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∈ wcel 2135 〈cop 3573 × cxp 4596 {coprab 5837 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 df-oprab 5840 |
This theorem is referenced by: (None) |
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