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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 6966 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Ref | Expression |
---|---|
dfrel4v | ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 6210 | . 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
2 | eqcom 2741 | . 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅) | |
3 | cnvcnv3 6209 | . . 3 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} | |
4 | 3 | eqeq2i 2747 | . 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 1536 class class class wbr 5147 {copab 5209 ◡ccnv 5687 Rel wrel 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 |
This theorem is referenced by: dfrel4 6212 dffn5 6966 fsplit 8140 pwsle 17538 tgphaus 24140 fneer 36335 inxp2 38348 dfxrn2 38357 1cosscnvxrn 38456 dfafn5a 47109 sprsymrelfo 47421 |
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