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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcoss4 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37748). (Contributed by Peter Mazsa, 12-Jul-2021.) |
Ref | Expression |
---|---|
dfcoss4 | ⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coss 37748 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
2 | rnxrn 37735 | . 2 ⊢ ran (𝑅 ⋉ 𝑅) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
3 | 1, 2 | eqtr4i 2762 | 1 ⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1780 class class class wbr 5148 {copab 5210 ran crn 5677 ⋉ cxrn 37509 ≀ ccoss 37510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7979 df-2nd 7980 df-ec 8711 df-xrn 37708 df-coss 37748 |
This theorem is referenced by: (None) |
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