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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 35664 | . 2 ⊢ ≀ 𝐴 = (𝐴 ∘ ◡𝐴) | |
2 | cnvexg 7631 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
3 | coexg 7636 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ◡𝐴 ∈ V) → (𝐴 ∘ ◡𝐴) ∈ V) | |
4 | 2, 3 | mpdan 685 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∘ ◡𝐴) ∈ V) |
5 | 1, 4 | eqeltrid 2919 | 1 ⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 ◡ccnv 5556 ∘ ccom 5561 ≀ ccoss 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-coss 35661 |
This theorem is referenced by: cosscnvex 35667 1cosscnvepresex 35668 1cossxrncnvepresex 35669 cosselrels 35738 elfunsALTVfunALTV 35932 |
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