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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 5567 | . . . . . 6 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
2 | 1 | sseli 3896 | . . . . 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 5120 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} |
6 | eleq1 2825 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
7 | opelxp 5587 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
8 | 6, 7 | bitrdi 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 634 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓))) |
11 | 10 | dfoprab3 7824 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} |
12 | 5, 11 | eqtri 2765 | 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 1543 ∈ wcel 2110 Vcvv 3408 〈cop 4547 {copab 5115 × cxp 5549 {coprab 7214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-iota 6338 df-fun 6382 df-fv 6388 df-oprab 7217 df-1st 7761 df-2nd 7762 |
This theorem is referenced by: dfoprab4f 7826 dfxp3 7831 xrninxp 36255 |
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