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Mirrors > Home > MPE Home > Th. List > dfrel4 | Structured version Visualization version GIF version |
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6821 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.) |
Ref | Expression |
---|---|
dfrel4.1 | ⊢ Ⅎ𝑥𝑅 |
dfrel4.2 | ⊢ Ⅎ𝑦𝑅 |
Ref | Expression |
---|---|
dfrel4 | ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel4v 6087 | . 2 ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎𝑅𝑏}) | |
2 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑥𝑎 | |
3 | dfrel4.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
4 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑥𝑏 | |
5 | 2, 3, 4 | nfbr 5121 | . . . 4 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
6 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑦𝑎 | |
7 | dfrel4.2 | . . . . 5 ⊢ Ⅎ𝑦𝑅 | |
8 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑦𝑏 | |
9 | 6, 7, 8 | nfbr 5121 | . . . 4 ⊢ Ⅎ𝑦 𝑎𝑅𝑏 |
10 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑎 𝑥𝑅𝑦 | |
11 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑏 𝑥𝑅𝑦 | |
12 | breq12 5079 | . . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎𝑅𝑏 ↔ 𝑥𝑅𝑦)) | |
13 | 5, 9, 10, 11, 12 | cbvopab 5146 | . . 3 ⊢ {〈𝑎, 𝑏〉 ∣ 𝑎𝑅𝑏} = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} |
14 | 13 | eqeq2i 2751 | . 2 ⊢ (𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎𝑅𝑏} ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
15 | 1, 14 | bitri 274 | 1 ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 Ⅎwnfc 2887 class class class wbr 5074 {copab 5136 Rel wrel 5590 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-br 5075 df-opab 5137 df-xp 5591 df-rel 5592 df-cnv 5593 |
This theorem is referenced by: feqmptdf 6832 |
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