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| Description: An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| errel | ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-er 8746 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
| 2 | 1 | simp1bi 1145 | 1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∪ cun 3948 ⊆ wss 3950 ◡ccnv 5683 dom cdm 5684 ∘ ccom 5688 Rel wrel 5689 Er wer 8743 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-er 8746 | 
| This theorem is referenced by: ercl 8757 ersym 8758 ertr 8761 ercnv 8767 erssxp 8769 erth 8797 iiner 8830 eqg0el 19202 frgpuplem 19791 qusxpid 33392 ismntop 34028 topfneec 36357 prter3 38884 | 
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