elcoeleqvrelsrel - Mathbox for Peter Mazsa

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Theorem elcoeleqvrelsrel 38581

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 38580	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (^◡ E ↾ 𝐴) ∈ EqvRels ))
2		1cosscnvepresex 38406	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (^◡ E ↾ 𝐴) ∈ V)
3		eleqvrelsrel 38579	. . . 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 38572	. 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 2109 Vcvv 3455 E cep 5545 ^◡ccnv 5645 ↾ cres 5648 ≀ ccoss 38166 EqvRels ceqvrels 38182 EqvRel weqvrel 38183 CoElEqvRels ccoeleqvrels 38184 CoElEqvRel wcoeleqvrel 38185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2928 df-ral 3047 df-rex 3056 df-rab 3412 df-v 3457 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-id 5541 df-eprel 5546 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-coss 38396 df-rels 38470 df-ssr 38483 df-refs 38495 df-refrels 38496 df-refrel 38497 df-syms 38527 df-symrels 38528 df-symrel 38529 df-trs 38557 df-trrels 38558 df-trrel 38559 df-eqvrels 38569 df-eqvrel 38570 df-coeleqvrels 38571 df-coeleqvrel 38572

This theorem is referenced by: (None)

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