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Mirrors > Home > MPE Home > Th. List > Mathboxes > petidres | Structured version Visualization version GIF version |
Description: A class is a partition by identity class restricted to it if and only if the cosets by the restricted identity class are in equivalence relation on it, cf. eqvrel1cossidres 38173. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
petidres | ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | petidres2 38201 | . 2 ⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴)) | |
2 | dfpart2 38152 | . 2 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴)) | |
3 | dferALTV2 38051 | . 2 ⊢ ( ≀ ( I ↾ 𝐴) ErALTV 𝐴 ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 303 | 1 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 I cid 5566 dom cdm 5669 ↾ cres 5671 / cqs 8704 ≀ ccoss 37556 EqvRel weqvrel 37573 ErALTV werALTV 37582 Disj wdisjALTV 37590 Part wpart 37595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rmo 3370 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 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 8707 df-qs 8711 df-coss 37794 df-refrel 37895 df-cnvrefrel 37910 df-symrel 37927 df-trrel 37957 df-eqvrel 37968 df-dmqs 38022 df-erALTV 38047 df-funALTV 38065 df-disjALTV 38088 df-part 38149 |
This theorem is referenced by: (None) |
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