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Mirrors > Home > MPE Home > Th. List > Mathboxes > petxrnidres2 | Structured version Visualization version GIF version |
Description: Class 𝐴 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. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
petxrnidres2 | ⊢ (( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjALTVxrnidres 38663 | . 2 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴)) | |
2 | 1 | petlemi 38718 | 1 ⊢ (( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 I cid 5596 dom cdm 5699 ↾ cres 5701 / cqs 8758 ⋉ cxrn 38083 ≀ ccoss 38084 EqvRel weqvrel 38101 Disj wdisjALTV 38118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fo 6578 df-fv 6580 df-1st 8026 df-2nd 8027 df-ec 8761 df-qs 8765 df-xrn 38276 df-coss 38316 df-refrel 38417 df-cnvrefrel 38432 df-symrel 38449 df-trrel 38479 df-eqvrel 38490 df-funALTV 38587 df-disjALTV 38610 |
This theorem is referenced by: petxrnidres 38728 |
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