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Mirrors > Home > ILE Home > Th. List > errel | GIF version |
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 6106 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
2 | 1 | simp1bi 919 | 1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∪ cun 2915 ⊆ wss 2917 ◡ccnv 4344 dom cdm 4345 ∘ ccom 4349 Rel wrel 4350 Er wer 6103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-er 6106 |
This theorem is referenced by: ercl 6117 ersym 6118 ertr 6121 ercnv 6127 erssxp 6129 erth 6150 iinerm 6178 |
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