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Mirrors > Home > MPE Home > Th. List > dfres2 | Structured version Visualization version GIF version |
Description: Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.) |
Ref | Expression |
---|---|
dfres2 | ⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5346 | . 2 ⊢ Rel (𝑅 ↾ 𝐴) | |
2 | relopab 5169 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
3 | vex 3176 | . . . . 5 ⊢ 𝑤 ∈ V | |
4 | 3 | brres 5323 | . . . 4 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧𝑅𝑤 ∧ 𝑧 ∈ 𝐴)) |
5 | df-br 4584 | . . . 4 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ 〈𝑧, 𝑤〉 ∈ (𝑅 ↾ 𝐴)) | |
6 | ancom 465 | . . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) | |
7 | 4, 5, 6 | 3bitr3i 289 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
8 | vex 3176 | . . . 4 ⊢ 𝑧 ∈ V | |
9 | eleq1 2676 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
10 | breq1 4586 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦)) | |
11 | 9, 10 | anbi12d 743 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) |
12 | breq2 4587 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤)) | |
13 | 12 | anbi2d 736 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) |
14 | 8, 3, 11, 13 | opelopab 4922 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
15 | 7, 14 | bitr4i 266 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ (𝑅 ↾ 𝐴) ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}) |
16 | 1, 2, 15 | eqrelriiv 5137 | 1 ⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 {copab 4642 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-res 5050 |
This theorem is referenced by: shftidt2 13669 |
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