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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 37546 and petlem 37551 via disjdmqseq 37544. (Contributed by Peter Mazsa, 16-Sep-2021.) |
Ref | Expression |
---|---|
disjdmqs | ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjdmqsss 37541 | . 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅)) | |
2 | disjdmqscossss 37542 | . 2 ⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅)) | |
3 | 1, 2 | eqssd 3996 | 1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 dom cdm 5670 / cqs 8687 ≀ ccoss 36912 Disj wdisjALTV 36946 |
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 5293 ax-nul 5300 ax-pr 5421 |
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 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-ec 8690 df-qs 8694 df-coss 37150 df-cnvrefrel 37266 df-disjALTV 37444 |
This theorem is referenced by: disjdmqseq 37544 |
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