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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 5991 | . 2 ⊢ Rel (𝑅 ↾ 𝐴) | |
| 2 | relopabv 5794 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
| 3 | vex 3458 | . . . 4 ⊢ 𝑧 ∈ V | |
| 4 | vex 3458 | . . . 4 ⊢ 𝑤 ∈ V | |
| 5 | eleq1w 2845 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
| 6 | breq1 5103 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦)) | |
| 7 | 5, 6 | anbi12d 641 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) |
| 8 | breq2 5104 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤)) | |
| 9 | 8 | anbi2d 639 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) |
| 10 | 3, 4, 7, 9 | opelopab 5513 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
| 11 | 4 | brresi 5974 | . . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
| 12 | df-br 5101 | . . 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 5762 | 1 ⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 399 = wceq 1560 ∈ wcel 2142 〈cop 4588 class class class wbr 5100 {copab 5162 ↾ cres 5649 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-ext 2734 ax-sep 5246 ax-pr 5390 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-sb 2091 df-clab 2741 df-cleq 2754 df-clel 2837 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-br 5101 df-opab 5163 df-xp 5653 df-rel 5654 df-res 5659 |
| This theorem is referenced by: shftidt2 15094 bj-imdiridlem 37674 dfres4 38795 cnvepres 38800 ressn2 39028 tfsconcatrev 43922 |
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