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| Mirrors > Home > MPE Home > Th. List > Mathboxes > inxprnres | Structured version Visualization version GIF version | ||
| Description: Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.) |
| Ref | Expression |
|---|---|
| inxprnres | ⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relinxp 5793 | . 2 ⊢ Rel (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) | |
| 2 | relopabv 5800 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
| 3 | eleq1w 2817 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
| 4 | breq1 5122 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦)) | |
| 5 | 3, 4 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) |
| 6 | breq2 5123 | . . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤)) | |
| 7 | 6 | anbi2d 630 | . . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) |
| 8 | 5, 7 | opelopabg 5513 | . . . 4 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) |
| 9 | 8 | el2v 3466 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
| 10 | brinxprnres 38255 | . . . 4 ⊢ (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) | |
| 11 | 10 | elv 3464 | . . 3 ⊢ (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
| 12 | df-br 5120 | . . 3 ⊢ (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ 〈𝑧, 𝑤〉 ∈ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))) | |
| 13 | 9, 11, 12 | 3bitr2ri 300 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}) |
| 14 | 1, 2, 13 | eqrelriiv 5769 | 1 ⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ∩ cin 3925 〈cop 4607 class class class wbr 5119 {copab 5181 × cxp 5652 ran crn 5655 ↾ cres 5656 |
| 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-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-br 5120 df-opab 5182 df-xp 5660 df-rel 5661 df-cnv 5662 df-dm 5664 df-rn 5665 df-res 5666 |
| This theorem is referenced by: dfres4 38257 |
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