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Mirrors > Home > MPE Home > Th. List > Mathboxes > petinidres | Structured version Visualization version GIF version |
Description: A class is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. Cf. br1cossinidres 37977, disjALTVinidres 38285 and eqvrel1cossinidres 38319. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
petinidres | ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | petinidres2 38348 | . 2 ⊢ (( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( I ↾ 𝐴)) / ≀ (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴)) | |
2 | dfpart2 38297 | . 2 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴)) | |
3 | dferALTV2 38196 | . 2 ⊢ ( ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( I ↾ 𝐴)) / ≀ (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 302 | 1 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∩ cin 3938 I cid 5569 dom cdm 5672 ↾ cres 5674 / cqs 8722 ≀ ccoss 37705 EqvRel weqvrel 37722 ErALTV werALTV 37731 Disj wdisjALTV 37739 Part wpart 37744 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rmo 3364 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8725 df-qs 8729 df-coss 37939 df-refrel 38040 df-cnvrefrel 38055 df-symrel 38072 df-trrel 38102 df-eqvrel 38113 df-dmqs 38167 df-erALTV 38192 df-funALTV 38210 df-disjALTV 38233 df-part 38294 |
This theorem is referenced by: (None) |
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