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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreldmqs | Structured version Visualization version GIF version |
Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.) |
Ref | Expression |
---|---|
eqvreldmqs | ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrel 37099 | . . 3 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
2 | 1 | bicomi 223 | . 2 ⊢ ( EqvRel ≀ (◡ E ↾ 𝐴) ↔ CoElEqvRel 𝐴) |
3 | dmqs1cosscnvepreseq 37174 | . 2 ⊢ ((dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴) | |
4 | 2, 3 | anbi12i 628 | 1 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∪ cuni 4869 E cep 5540 ◡ccnv 5636 dom cdm 5637 ↾ cres 5639 / cqs 8653 ≀ ccoss 36684 ∼ ccoels 36685 EqvRel weqvrel 36701 CoElEqvRel wcoeleqvrel 36703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-eprel 5541 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ec 8656 df-qs 8660 df-coss 36923 df-coels 36924 df-coeleqvrel 37099 |
This theorem is referenced by: mpet3 37348 |
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