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Mirrors > Home > ILE Home > Th. List > dfoprab4 | 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 4695 | . . . . . 6 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
2 | 1 | sseli 3124 | . . . . 5 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 ∈ (V × V)) |
3 | 2 | adantr 274 | . . . 4 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) → 𝑤 ∈ (V × V)) |
4 | 3 | pm4.71ri 390 | . . 3 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))) |
5 | 4 | opabbii 4032 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} |
6 | eleq1 2220 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
7 | opelxp 4617 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
8 | 6, 7 | bitrdi 195 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
9 | dfoprab4.1 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | anbi12d 465 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓))) |
11 | 10 | dfoprab3 6140 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} |
12 | 5, 11 | eqtri 2178 | 1 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 Vcvv 2712 〈cop 3563 {copab 4025 × cxp 4585 {coprab 5826 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fo 5177 df-fv 5179 df-oprab 5829 df-1st 6089 df-2nd 6090 |
This theorem is referenced by: dfoprab4f 6142 dfxp3 6143 |
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