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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 38183. (Contributed by Peter Mazsa, 25-Dec-2024.) |
Ref | Expression |
---|---|
partimeq | ⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossex 37923 | . 2 ⊢ (𝑅 ∈ 𝑉 → ≀ 𝑅 ∈ V) | |
2 | partim 38312 | . 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | |
3 | erimeq 38183 | . 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 1533 ∈ wcel 2098 Vcvv 3473 ≀ ccoss 37681 ∼ ccoels 37682 ErALTV werALTV 37707 Part wpart 37720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-id 5580 df-eprel 5586 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ec 8733 df-qs 8737 df-coss 37915 df-coels 37916 df-refrel 38016 df-cnvrefrel 38031 df-symrel 38048 df-trrel 38078 df-eqvrel 38089 df-dmqs 38143 df-erALTV 38168 df-disjALTV 38209 df-part 38270 |
This theorem is referenced by: (None) |
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