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Mirrors > Home > MPE Home > Th. List > Mathboxes > elcoeleqvrelsrel | Structured version Visualization version GIF version |
Description: For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate. (Contributed by Peter Mazsa, 24-Jul-2023.) |
Ref | Expression |
---|---|
elcoeleqvrelsrel | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elcoeleqvrels 37781 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | |
2 | 1cosscnvepresex 37607 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | |
3 | eleqvrelsrel 37780 | . . . 4 ⊢ ( ≀ (◡ E ↾ 𝐴) ∈ V → ( ≀ (◡ E ↾ 𝐴) ∈ EqvRels ↔ EqvRel ≀ (◡ E ↾ 𝐴))) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ( ≀ (◡ E ↾ 𝐴) ∈ EqvRels ↔ EqvRel ≀ (◡ E ↾ 𝐴))) |
5 | 1, 4 | bitrd 279 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ EqvRel ≀ (◡ E ↾ 𝐴))) |
6 | df-coeleqvrel 37773 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
7 | 5, 6 | bitr4di 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2105 Vcvv 3473 E cep 5579 ◡ccnv 5675 ↾ cres 5678 ≀ ccoss 37359 EqvRels ceqvrels 37375 EqvRel weqvrel 37376 CoElEqvRels ccoeleqvrels 37377 CoElEqvRel wcoeleqvrel 37378 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-coss 37597 df-rels 37671 df-ssr 37684 df-refs 37696 df-refrels 37697 df-refrel 37698 df-syms 37728 df-symrels 37729 df-symrel 37730 df-trs 37758 df-trrels 37759 df-trrel 37760 df-eqvrels 37770 df-eqvrel 37771 df-coeleqvrels 37772 df-coeleqvrel 37773 |
This theorem is referenced by: (None) |
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