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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 4768 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | relin1 4759 | . . 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 4768 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} | |
5 | sban 1967 | . . . 4 ⊢ ([𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓) ↔ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓)) | |
6 | sban 1967 | . . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓)) | |
7 | 6 | sbbii 1776 | . . . 4 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓)) |
8 | opelopabsbALT 4274 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) | |
9 | opelopabsbALT 4274 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓) | |
10 | 8, 9 | anbi12i 460 | . . . 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 213 | . . 3 ⊢ ((〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓)) |
12 | elin 3333 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) | |
13 | opelopabsbALT 4274 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓)) | |
14 | 11, 12, 13 | 3bitr4i 212 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)}) |
15 | 3, 4, 14 | eqrelriiv 4735 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 [wsb 1773 ∈ wcel 2160 ∩ cin 3143 〈cop 3610 {copab 4078 Rel wrel 4646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-opab 4080 df-xp 4647 df-rel 4648 |
This theorem is referenced by: inxp 4776 resopab 4966 cnvin 5051 fndmin 5639 enq0enq 7448 |
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