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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 6023 | . 2 ⊢ Rel (𝑅 ↾ 𝐴) | |
| 2 | relopabv 5831 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
| 3 | vex 3484 | . . . 4 ⊢ 𝑧 ∈ V | |
| 4 | vex 3484 | . . . 4 ⊢ 𝑤 ∈ V | |
| 5 | eleq1w 2824 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
| 6 | breq1 5146 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦)) | |
| 7 | 5, 6 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) |
| 8 | breq2 5147 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤)) | |
| 9 | 8 | anbi2d 630 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) |
| 10 | 3, 4, 7, 9 | opelopab 5547 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
| 11 | 4 | brresi 6006 | . . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
| 12 | df-br 5144 | . . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ 〈𝑧, 𝑤〉 ∈ (𝑅 ↾ 𝐴)) | |
| 13 | 10, 11, 12 | 3bitr2ri 300 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ (𝑅 ↾ 𝐴) ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}) |
| 14 | 1, 2, 13 | eqrelriiv 5800 | 1 ⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 {copab 5205 ↾ cres 5687 |
| 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 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 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-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-res 5697 |
| This theorem is referenced by: shftidt2 15120 bj-imdiridlem 37186 dfres4 38294 cnvepres 38299 ressn2 38443 tfsconcatrev 43361 |
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