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Mirrors > Home > MPE Home > Th. List > errel | Structured version Visualization version 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 8278 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
2 | 1 | simp1bi 1137 | 1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∪ cun 3931 ⊆ wss 3933 ◡ccnv 5547 dom cdm 5548 ∘ ccom 5552 Rel wrel 5553 Er wer 8275 |
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 208 df-an 397 df-3an 1081 df-er 8278 |
This theorem is referenced by: ercl 8289 ersym 8290 ertr 8293 ercnv 8299 erssxp 8301 erth 8327 iiner 8358 frgpuplem 18827 eqg0el 30853 qusxpid 30855 ismntop 31166 topfneec 33600 prter3 35898 |
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