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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 38635. (Contributed by Peter Mazsa, 25-Dec-2024.) |
Ref | Expression |
---|---|
partimeq | ⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossex 38375 | . 2 ⊢ (𝑅 ∈ 𝑉 → ≀ 𝑅 ∈ V) | |
2 | partim 38764 | . 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | |
3 | erimeq 38635 | . 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 2108 Vcvv 3488 ≀ ccoss 38135 ∼ ccoels 38136 ErALTV werALTV 38161 Part wpart 38174 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-id 5593 df-eprel 5599 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ec 8765 df-qs 8769 df-coss 38367 df-coels 38368 df-refrel 38468 df-cnvrefrel 38483 df-symrel 38500 df-trrel 38530 df-eqvrel 38541 df-dmqs 38595 df-erALTV 38620 df-disjALTV 38661 df-part 38722 |
This theorem is referenced by: (None) |
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