elcoeleqvrelsrel - Mathbox for Peter Mazsa

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Theorem elcoeleqvrelsrel 38587

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 38586	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (^◡ E ↾ 𝐴) ∈ EqvRels ))
2		1cosscnvepresex 38412	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (^◡ E ↾ 𝐴) ∈ V)
3		eleqvrelsrel 38585	. . . 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 38578	. 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 3447 E cep 5537 ^◡ccnv 5637 ↾ cres 5640 ≀ ccoss 38169 EqvRels ceqvrels 38185 EqvRel weqvrel 38186 CoElEqvRels ccoeleqvrels 38187 CoElEqvRel wcoeleqvrel 38188

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

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 2533 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-id 5533 df-eprel 5538 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-coss 38402 df-rels 38476 df-ssr 38489 df-refs 38501 df-refrels 38502 df-refrel 38503 df-syms 38533 df-symrels 38534 df-symrel 38535 df-trs 38563 df-trrels 38564 df-trrel 38565 df-eqvrels 38575 df-eqvrel 38576 df-coeleqvrels 38577 df-coeleqvrel 38578

This theorem is referenced by: (None)

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