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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 6011 | . 2 ⊢ Rel (𝑅 ↾ 𝐴) | |
2 | relopabv 5822 | . 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
3 | vex 3479 | . . . 4 ⊢ 𝑧 ∈ V | |
4 | vex 3479 | . . . 4 ⊢ 𝑤 ∈ V | |
5 | eleq1w 2817 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
6 | breq1 5152 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦)) | |
7 | 5, 6 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) |
8 | breq2 5153 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤)) | |
9 | 8 | anbi2d 630 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))) |
10 | 3, 4, 7, 9 | opelopab 5543 | . . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
11 | 4 | brresi 5991 | . . 3 ⊢ (𝑧(𝑅 ↾ 𝐴)𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)) |
12 | df-br 5150 | . . 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 5791 | 1 ⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4635 class class class wbr 5149 {copab 5211 ↾ cres 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-res 5689 |
This theorem is referenced by: shftidt2 15028 bj-imdiridlem 36114 dfres4 37210 cnvepres 37215 ressn2 37360 tfsconcatrev 42146 |
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