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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 6967 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) |
| Ref | Expression |
|---|---|
| dfrel4v | ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfrel2 6209 | . 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
| 2 | eqcom 2744 | . 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅) | |
| 3 | cnvcnv3 6208 | . . 3 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} | |
| 4 | 3 | eqeq2i 2750 | . 2 ⊢ (𝑅 = ◡◡𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
| 5 | 1, 2, 4 | 3bitri 297 | 1 ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 class class class wbr 5143 {copab 5205 ◡ccnv 5684 Rel wrel 5690 |
| 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-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-cnv 5693 |
| This theorem is referenced by: dfrel4 6211 dffn5 6967 fsplit 8142 pwsle 17537 tgphaus 24125 fneer 36354 inxp2 38368 dfxrn2 38377 1cosscnvxrn 38476 dfafn5a 47172 sprsymrelfo 47484 |
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