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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 6464 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Ref | Expression |
---|---|
dfrel4v | ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 5798 | . 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
2 | eqcom 2804 | . 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅) | |
3 | cnvcnv3 5797 | . . 3 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} | |
4 | 3 | eqeq2i 2809 | . 2 ⊢ (𝑅 = ◡◡𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
5 | 1, 2, 4 | 3bitri 289 | 1 ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1653 class class class wbr 4841 {copab 4903 ◡ccnv 5309 Rel wrel 5315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-opab 4904 df-xp 5316 df-rel 5317 df-cnv 5318 |
This theorem is referenced by: dfrel4 5800 dffn5 6464 fsplit 7517 pwsle 16464 tgphaus 22245 fneer 32852 inxp2 34615 dfxrn2 34624 1cosscnvxrn 34711 dfafn5a 42002 sprsymrelfo 42534 |
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