elcoeleqvrelsrel - Mathbox for Peter Mazsa

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Theorem elcoeleqvrelsrel 37454

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.)

**Assertion**
Ref	Expression
elcoeleqvrelsrel	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴))

**Proof of Theorem elcoeleqvrelsrel**
Step	Hyp	Ref	Expression
1		elcoeleqvrels 37453	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (^◡ E ↾ 𝐴) ∈ EqvRels ))
2		1cosscnvepresex 37279	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (^◡ E ↾ 𝐴) ∈ V)
3		eleqvrelsrel 37452	. . . 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 278	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ EqvRel ≀ (^◡ E ↾ 𝐴)))
6		df-coeleqvrel 37445	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (^◡ E ↾ 𝐴))
7	5, 6	bitr4di 288	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 Vcvv 3474 E cep 5578 ^◡ccnv 5674 ↾ cres 5677 ≀ ccoss 37031 EqvRels ceqvrels 37047 EqvRel weqvrel 37048 CoElEqvRels ccoeleqvrels 37049 CoElEqvRel wcoeleqvrel 37050

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-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

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-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-id 5573 df-eprel 5579 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-coss 37269 df-rels 37343 df-ssr 37356 df-refs 37368 df-refrels 37369 df-refrel 37370 df-syms 37400 df-symrels 37401 df-symrel 37402 df-trs 37430 df-trrels 37431 df-trrel 37432 df-eqvrels 37442 df-eqvrel 37443 df-coeleqvrels 37444 df-coeleqvrel 37445

This theorem is referenced by: (None)

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