elcoeleqvrelsrel - Mathbox for Peter Mazsa

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Theorem elcoeleqvrelsrel 36709

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 36708	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (^◡ E ↾ 𝐴) ∈ EqvRels ))
2		1cosscnvepresex 36544	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (^◡ E ↾ 𝐴) ∈ V)
3		eleqvrelsrel 36707	. . . 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 36700	. 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 2106 Vcvv 3432 E cep 5494 ^◡ccnv 5588 ↾ cres 5591 ≀ ccoss 36333 EqvRels ceqvrels 36349 EqvRel weqvrel 36350 CoElEqvRels ccoeleqvrels 36351 CoElEqvRel wcoeleqvrel 36352

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-id 5489 df-eprel 5495 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-coss 36537 df-rels 36603 df-ssr 36616 df-refs 36628 df-refrels 36629 df-refrel 36630 df-syms 36656 df-symrels 36657 df-symrel 36658 df-trs 36686 df-trrels 36687 df-trrel 36688 df-eqvrels 36697 df-eqvrel 36698 df-coeleqvrels 36699 df-coeleqvrel 36700

This theorem is referenced by: (None)

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