|   | Mathbox for Peter Mazsa | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > partim | Structured version Visualization version GIF version | ||
| Description: Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 38808. (Contributed by Peter Mazsa, 17-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| partim | ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | partim2 38808 | . 2 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) | |
| 2 | dfpart2 38770 | . 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
| 3 | dferALTV2 38669 | . 2 ⊢ ( ≀ 𝑅 ErALTV 𝐴 ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) | |
| 4 | 1, 2, 3 | 3imtr4i 292 | 1 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 dom cdm 5685 / cqs 8744 ≀ ccoss 38182 EqvRel weqvrel 38199 ErALTV werALTV 38208 Disj wdisjALTV 38216 Part wpart 38221 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 df-coss 38412 df-refrel 38513 df-cnvrefrel 38528 df-symrel 38545 df-trrel 38575 df-eqvrel 38586 df-dmqs 38640 df-erALTV 38665 df-disjALTV 38706 df-part 38767 | 
| This theorem is referenced by: partimeq 38810 partimcomember 38836 | 
| Copyright terms: Public domain | W3C validator |