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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 38577 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | |
2 | 1cosscnvepresex 38403 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | |
3 | eleqvrelsrel 38576 | . . . 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 38569 | . 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 206 ∈ wcel 2106 Vcvv 3478 E cep 5588 ◡ccnv 5688 ↾ cres 5691 ≀ ccoss 38162 EqvRels ceqvrels 38178 EqvRel weqvrel 38179 CoElEqvRels ccoeleqvrels 38180 CoElEqvRel wcoeleqvrel 38181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-coss 38393 df-rels 38467 df-ssr 38480 df-refs 38492 df-refrels 38493 df-refrel 38494 df-syms 38524 df-symrels 38525 df-symrel 38526 df-trs 38554 df-trrels 38555 df-trrel 38556 df-eqvrels 38566 df-eqvrel 38567 df-coeleqvrels 38568 df-coeleqvrel 38569 |
This theorem is referenced by: (None) |
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