Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6397 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
2 | 1 | simp1bi 981 | 1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∪ cun 3039 ⊆ wss 3041 ◡ccnv 4508 dom cdm 4509 ∘ ccom 4513 Rel wrel 4514 Er wer 6394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-er 6397 |
This theorem is referenced by: ercl 6408 ersym 6409 ertr 6412 ercnv 6418 erssxp 6420 erth 6441 iinerm 6469 |
Copyright terms: Public domain | W3C validator |