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| Mirrors > Home > MPE Home > Th. List > Mathboxes > partimeq | Structured version Visualization version GIF version | ||
| Description: Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 38680. (Contributed by Peter Mazsa, 25-Dec-2024.) |
| Ref | Expression |
|---|---|
| partimeq | ⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cossex 38420 | . 2 ⊢ (𝑅 ∈ 𝑉 → ≀ 𝑅 ∈ V) | |
| 2 | partim 38809 | . 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | |
| 3 | erimeq 38680 | . 2 ⊢ ( ≀ 𝑅 ∈ V → ( ≀ 𝑅 ErALTV 𝐴 → ∼ 𝐴 = ≀ 𝑅)) | |
| 4 | 1, 2, 3 | syl2im 40 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ≀ ccoss 38182 ∼ ccoels 38183 ErALTV werALTV 38208 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-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-ne 2941 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-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-id 5578 df-eprel 5584 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-coels 38413 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: (None) |
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