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Mirrors > Home > MPE Home > Th. List > dfrel4v | Structured version Visualization version GIF version |
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6241 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Ref | Expression |
---|---|
dfrel4v | ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 5583 | . 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
2 | eqcom 2629 | . 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅) | |
3 | cnvcnv3 5582 | . . 3 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} | |
4 | 3 | eqeq2i 2634 | . 2 ⊢ (𝑅 = ◡◡𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
5 | 1, 2, 4 | 3bitri 286 | 1 ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 class class class wbr 4653 {copab 4712 ◡ccnv 5113 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: dfrel4 5585 dffn5 6241 fsplit 7282 pwsle 16152 tgphaus 21920 fneer 32348 inxp2 34129 dfafn5a 41240 sprsymrelfo 41747 |
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