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Mirrors > Home > MPE Home > Th. List > Mathboxes > petxrnidres | Structured version Visualization version GIF version |
Description: A class is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. Cf. br1cossxrnidres 37978, disjALTVxrnidres 38285 and eqvrel1cossxrnidres 38319. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
petxrnidres | ⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | petxrnidres2 38349 | . 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 38296 | . 2 ⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴)) | |
3 | dferALTV2 38195 | . 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 I cid 5569 dom cdm 5672 ↾ cres 5674 / cqs 8720 ⋉ cxrn 37703 ≀ ccoss 37704 EqvRel weqvrel 37721 ErALTV werALTV 37730 Disj wdisjALTV 37738 Part wpart 37743 |
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 ax-un 7737 |
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-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 df-1st 7989 df-2nd 7990 df-ec 8723 df-qs 8727 df-xrn 37898 df-coss 37938 df-refrel 38039 df-cnvrefrel 38054 df-symrel 38071 df-trrel 38101 df-eqvrel 38112 df-dmqs 38166 df-erALTV 38191 df-funALTV 38209 df-disjALTV 38232 df-part 38293 |
This theorem is referenced by: (None) |
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