![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > disjdmqs | Structured version Visualization version GIF version |
Description: If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 38311 and petlem 38316 via disjdmqseq 38309. (Contributed by Peter Mazsa, 16-Sep-2021.) |
Ref | Expression |
---|---|
disjdmqs | ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjdmqsss 38306 | . 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅)) | |
2 | disjdmqscossss 38307 | . 2 ⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅)) | |
3 | 1, 2 | eqssd 3999 | 1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 dom cdm 5682 / cqs 8730 ≀ ccoss 37681 Disj wdisjALTV 37715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rmo 3374 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ec 8733 df-qs 8737 df-coss 37915 df-cnvrefrel 38031 df-disjALTV 38209 |
This theorem is referenced by: disjdmqseq 38309 |
Copyright terms: Public domain | W3C validator |