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Mirrors > Home > MPE Home > Th. List > Mathboxes > detinidres | Structured version Visualization version GIF version |
Description: The cosets by the intersection with the restricted identity relation are in equivalence relation if and only if the intersection with the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
detinidres | ⊢ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjALTVinidres 37533 | . 2 ⊢ Disj (𝑅 ∩ ( I ↾ 𝐴)) | |
2 | 1 | detlem 37559 | 1 ⊢ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∩ cin 3945 I cid 5569 ↾ cres 5674 ≀ ccoss 36949 EqvRel weqvrel 36966 Disj wdisjALTV 36983 |
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 5295 ax-nul 5302 ax-pr 5423 |
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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 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-coss 37187 df-refrel 37288 df-cnvrefrel 37303 df-symrel 37320 df-trrel 37350 df-eqvrel 37361 df-funALTV 37458 df-disjALTV 37481 |
This theorem is referenced by: (None) |
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