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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmqscoelseq | Structured version Visualization version GIF version |
Description: Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.) |
Ref | Expression |
---|---|
dmqscoelseq | ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmcoels 35731 | . . 3 ⊢ dom ∼ 𝐴 = ∪ 𝐴 | |
2 | 1 | qseq1i 35580 | . 2 ⊢ (dom ∼ 𝐴 / ∼ 𝐴) = (∪ 𝐴 / ∼ 𝐴) |
3 | 2 | eqeq1i 2825 | 1 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ∪ cuni 4831 dom cdm 5548 / cqs 8281 ∼ ccoels 35488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-qs 8288 df-coss 35693 df-coels 35694 |
This theorem is referenced by: dmqs1cosscnvepreseq 35930 dfmember3 35942 |
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