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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjdmqseq | Structured version Visualization version GIF version |
Description: If a relation is disjoint, its domain quotient is equal to a class if and only if the domain quotient of the cosets by it is equal to the class. General version of eldisjn0el 37481 (which is the closest theorem to the former prter2 37556). Lemma for partim2 37482 and petlem 37487. (Contributed by Peter Mazsa, 16-Sep-2021.) |
Ref | Expression |
---|---|
disjdmqseq | ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjdmqs 37479 | . 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅)) | |
2 | 1 | eqeq1d 2733 | 1 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 dom cdm 5669 / cqs 8685 ≀ ccoss 36848 Disj wdisjALTV 36882 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rmo 3375 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8688 df-qs 8692 df-coss 37086 df-cnvrefrel 37202 df-disjALTV 37380 |
This theorem is referenced by: eldisjn0el 37481 partim2 37482 petlem 37487 |
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