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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 37672 and petlem 37677 via disjdmqseq 37670. (Contributed by Peter Mazsa, 16-Sep-2021.) |
Ref | Expression |
---|---|
disjdmqs | ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjdmqsss 37667 | . 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅)) | |
2 | disjdmqscossss 37668 | . 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 1541 dom cdm 5676 / cqs 8701 ≀ ccoss 37038 Disj wdisjALTV 37072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rmo 3376 df-rab 3433 df-v 3476 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-br 5149 df-opab 5211 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-ec 8704 df-qs 8708 df-coss 37276 df-cnvrefrel 37392 df-disjALTV 37570 |
This theorem is referenced by: disjdmqseq 37670 |
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