eleqvrelsrel - Mathbox for Peter Mazsa

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Theorem eleqvrelsrel 35862

Description: For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate. (Contributed by Peter Mazsa, 24-Aug-2021.)

**Assertion**
Ref	Expression
eleqvrelsrel	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅))

**Proof of Theorem eleqvrelsrel**
Step	Hyp	Ref	Expression
1		elrelsrel 35760	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2	1	anbi2d 630	. 2 ⊢ (𝑅 ∈ 𝑉 → (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ^◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ^◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅)))
3		eleqvrels2 35860	. 2 ⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ^◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
4		dfeqvrel2 35858	. 2 ⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ^◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅))
5	2, 3, 4	3bitr4g 316	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 ∈ wcel 2113 ⊆ wss 3929 I cid 5452 ^◡ccnv 5547 dom cdm 5548 ↾ cres 5550 ∘ ccom 5552 Rel wrel 5553 Rels crels 35488 EqvRels ceqvrels 35502 EqvRel weqvrel 35503

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-rels 35758 df-ssr 35771 df-refs 35783 df-refrels 35784 df-refrel 35785 df-syms 35811 df-symrels 35812 df-symrel 35813 df-trs 35841 df-trrels 35842 df-trrel 35843 df-eqvrels 35852 df-eqvrel 35853

This theorem is referenced by: elcoeleqvrelsrel 35864 brerser 35943

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