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Mirrors > Home > MPE Home > Th. List > Mathboxes > detxrnidres | Structured version Visualization version GIF version |
Description: The cosets by the range Cartesian product with the restricted identity relation are in equivalence relation if and only if the range Cartesian product with the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
detxrnidres | ⊢ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjALTVxrnidres 38285 | . 2 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴)) | |
2 | 1 | detlem 38310 | 1 ⊢ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 I cid 5569 ↾ cres 5674 ⋉ cxrn 37703 ≀ ccoss 37704 EqvRel weqvrel 37721 Disj wdisjALTV 37738 |
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-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-xrn 37898 df-coss 37938 df-refrel 38039 df-cnvrefrel 38054 df-symrel 38071 df-trrel 38101 df-eqvrel 38112 df-funALTV 38209 df-disjALTV 38232 |
This theorem is referenced by: (None) |
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