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Mirrors > Home > ILE Home > Th. List > inopab | GIF version |
Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
inopab | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 4666 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | relin1 4657 | . . 3 ⊢ (Rel {〈𝑥, 𝑦〉 ∣ 𝜑} → Rel ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Rel ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
4 | relopab 4666 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} | |
5 | sban 1928 | . . . 4 ⊢ ([𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓) ↔ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓)) | |
6 | sban 1928 | . . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓)) | |
7 | 6 | sbbii 1738 | . . . 4 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓)) |
8 | opelopabsbALT 4181 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) | |
9 | opelopabsbALT 4181 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓) | |
10 | 8, 9 | anbi12i 455 | . . . 4 ⊢ ((〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓)) |
11 | 5, 7, 10 | 3bitr4ri 212 | . . 3 ⊢ ((〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓)) |
12 | elin 3259 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) | |
13 | opelopabsbALT 4181 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓)) | |
14 | 11, 12, 13 | 3bitr4i 211 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)}) |
15 | 3, 4, 14 | eqrelriiv 4633 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 [wsb 1735 ∩ cin 3070 〈cop 3530 {copab 3988 Rel wrel 4544 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 df-rel 4546 |
This theorem is referenced by: inxp 4673 resopab 4863 cnvin 4946 fndmin 5527 enq0enq 7239 |
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