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Mirrors > Home > MPE Home > Th. List > oprab4 | Structured version Visualization version 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 5585 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | 1 | anbi1i 625 | . 2 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
3 | 2 | oprabbii 7215 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4566 × cxp 5547 {coprab 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5121 df-xp 5555 df-oprab 7154 |
This theorem is referenced by: (None) |
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