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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 5884 | . 2 ⊢ Rel (𝑅 ↾ 𝐴) | |
2 | relopab 5698 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
3 | vex 3499 | . . . 4 ⊢ 𝑧 ∈ V | |
4 | vex 3499 | . . . 4 ⊢ 𝑤 ∈ V | |
5 | eleq1w 2897 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
6 | breq1 5071 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦)) | |
7 | 5, 6 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) |
8 | breq2 5072 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤)) | |
9 | 8 | anbi2d 630 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) |
10 | 3, 4, 7, 9 | opelopab 5431 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
11 | 4 | brresi 5864 | . . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
12 | df-br 5069 | . . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ 〈𝑧, 𝑤〉 ∈ (𝑅 ↾ 𝐴)) | |
13 | 10, 11, 12 | 3bitr2ri 302 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ (𝑅 ↾ 𝐴) ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}) |
14 | 1, 2, 13 | eqrelriiv 5665 | 1 ⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 {copab 5130 ↾ cres 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-res 5569 |
This theorem is referenced by: shftidt2 14442 bj-imdirid 34477 dfres4 35552 cnvepres 35557 |
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