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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 6534 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
2 | 1 | simp2bi 1013 | 1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∪ cun 3127 ⊆ wss 3129 ◡ccnv 4625 dom cdm 4626 ∘ ccom 4630 Rel wrel 4631 Er wer 6531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-er 6534 |
This theorem is referenced by: ercl 6545 erref 6554 errn 6556 erssxp 6557 erexb 6559 ereldm 6577 uniqs2 6594 iinerm 6606 th3qlem1 6636 0nnq 7362 nnnq0lem1 7444 prsrlem1 7740 gt0srpr 7746 0nsr 7747 |
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