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| Mirrors > Home > MPE Home > Th. List > inopab | Structured version Visualization version 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 | relopabv 5831 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | relin1 5822 | . . 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 | relopabv 5831 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} | |
| 5 | sban 2080 | . . . 4 ⊢ ([𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦]𝜓) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)) | |
| 6 | sban 2080 | . . . . 5 ⊢ ([𝑤 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦]𝜓)) | |
| 7 | 6 | sbbii 2076 | . . . 4 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦]𝜓)) |
| 8 | vopelopabsb 5534 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | |
| 9 | vopelopabsb 5534 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓) | |
| 10 | 8, 9 | anbi12i 628 | . . . 4 ⊢ ((〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)) |
| 11 | 5, 7, 10 | 3bitr4ri 304 | . . 3 ⊢ ((〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ 𝜓)) |
| 12 | elin 3967 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) | |
| 13 | vopelopabsb 5534 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ 𝜓)) | |
| 14 | 11, 12, 13 | 3bitr4i 303 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)}) |
| 15 | 3, 4, 14 | eqrelriiv 5800 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∩ {〈𝑥, 𝑦〉 ∣ 𝜓}) = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝜓)} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 [wsb 2064 ∈ wcel 2108 ∩ cin 3950 〈cop 4632 {copab 5205 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: inxpOLD 5843 resopab 6052 fndmin 7065 cnvoprab 8085 epinid0 9640 cnvepnep 9648 wemapwe 9737 dfiso2 17816 frgpuplem 19790 pjfval2 21729 ltbwe 22062 opsrtoslem1 22079 lgsquadlem3 27426 disjecxrn 38390 br1cosscnvxrn 38475 1cosscnvxrn 38476 dnwech 43060 fgraphopab 43215 |
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