Theorem erdm 6790

Description: The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
erdm	⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Proof of Theorem erdm

Step	Hyp	Ref	Expression
1		df-er 6780	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1040	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1398   ∪ cun 3212   ⊆ wss 3214  ^◡ccnv 4753  dom cdm 4754   ∘ ccom 4758  Rel wrel 4759   Er wer 6777

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 1007  df-er 6780

This theorem is referenced by:  ercl  6791  erref  6800  errn  6802  erssxp  6803  erexb  6805  ereldm  6825  uniqs2  6842  iinerm  6854  th3qlem1  6884  0nnq  7695  nnnq0lem1  7777  prsrlem1  8073  gt0srpr  8079  0nsr  8080  divsfval  13592