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Mirrors > Home > MPE Home > Th. List > Mathboxes > cossex | Structured version Visualization version GIF version |
Description: If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.) |
Ref | Expression |
---|---|
cossex | ⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcoss3 36865 | . 2 ⊢ ≀ 𝐴 = (𝐴 ∘ ◡𝐴) | |
2 | cnvexg 7858 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
3 | coexg 7863 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ◡𝐴 ∈ V) → (𝐴 ∘ ◡𝐴) ∈ V) | |
4 | 2, 3 | mpdan 685 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∘ ◡𝐴) ∈ V) |
5 | 1, 4 | eqeltrid 2842 | 1 ⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3444 ◡ccnv 5631 ∘ ccom 5636 ≀ ccoss 36623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-coss 36862 |
This theorem is referenced by: cosscnvex 36871 1cosscnvepresex 36872 1cossxrncnvepresex 36873 cosselrels 36947 elfunsALTVfunALTV 37148 partimeq 37260 |
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