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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 38787 (which is the closest theorem to the former prter2 38862). Lemma for partim2 38788 and petlem 38793. (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 38785 | . 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅)) | |
2 | 1 | eqeq1d 2736 | 1 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 dom cdm 5688 / cqs 8742 ≀ ccoss 38161 Disj wdisjALTV 38195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rmo 3377 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ec 8745 df-qs 8749 df-coss 38392 df-cnvrefrel 38508 df-disjALTV 38686 |
This theorem is referenced by: eldisjn0el 38787 partim2 38788 petlem 38793 |
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