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Mirrors > Home > ILE Home > Th. List > erdm | GIF version |
Description: The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
erdm | ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-er 6492 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
2 | 1 | simp2bi 1002 | 1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∪ cun 3109 ⊆ wss 3111 ◡ccnv 4597 dom cdm 4598 ∘ ccom 4602 Rel wrel 4603 Er wer 6489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-er 6492 |
This theorem is referenced by: ercl 6503 erref 6512 errn 6514 erssxp 6515 erexb 6517 ereldm 6535 uniqs2 6552 iinerm 6564 th3qlem1 6594 0nnq 7296 nnnq0lem1 7378 prsrlem1 7674 gt0srpr 7680 0nsr 7681 |
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