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Mirrors > Home > MPE Home > Th. List > Mathboxes > eleqvrelsrel | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
eleqvrelsrel | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrelsrel 36978 | . . 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 37083 | . 2 ⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) | |
4 | dfeqvrel2 37081 | . 2 ⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅)) | |
5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3915 I cid 5535 ◡ccnv 5637 dom cdm 5638 ↾ cres 5640 ∘ ccom 5642 Rel wrel 5643 Rels crels 36665 EqvRels ceqvrels 36679 EqvRel weqvrel 36680 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-rels 36976 df-ssr 36989 df-refs 37001 df-refrels 37002 df-refrel 37003 df-syms 37033 df-symrels 37034 df-symrel 37035 df-trs 37063 df-trrels 37064 df-trrel 37065 df-eqvrels 37075 df-eqvrel 37076 |
This theorem is referenced by: elcoeleqvrelsrel 37087 brerser 37168 |
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