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Mirrors > Home > MPE Home > Th. List > dfoprab4 | Structured version Visualization version GIF version |
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfoprab4.1 | ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dfoprab4 | ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 5535 | . . . . . 6 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
2 | 1 | sseli 3911 | . . . . 5 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 ∈ (V × V)) |
3 | 2 | adantr 484 | . . . 4 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) → 𝑤 ∈ (V × V)) |
4 | 3 | pm4.71ri 564 | . . 3 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))) |
5 | 4 | opabbii 5097 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} |
6 | eleq1 2877 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
7 | opelxp 5555 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
8 | 6, 7 | syl6bb 290 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
9 | dfoprab4.1 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | anbi12d 633 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓))) |
11 | 10 | dfoprab3 7734 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} |
12 | 5, 11 | eqtri 2821 | 1 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 {copab 5092 × cxp 5517 {coprab 7136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-oprab 7139 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: dfoprab4f 7736 dfxp3 7741 xrninxp 35800 |
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