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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 38661. (Contributed by Peter Mazsa, 25-Dec-2024.) |
Ref | Expression |
---|---|
partimeq | ⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossex 38401 | . 2 ⊢ (𝑅 ∈ 𝑉 → ≀ 𝑅 ∈ V) | |
2 | partim 38790 | . 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | |
3 | erimeq 38661 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ≀ ccoss 38162 ∼ ccoels 38163 ErALTV werALTV 38188 Part wpart 38201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 df-coss 38393 df-coels 38394 df-refrel 38494 df-cnvrefrel 38509 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 df-dmqs 38621 df-erALTV 38646 df-disjALTV 38687 df-part 38748 |
This theorem is referenced by: (None) |
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