Theorem erdm 6777

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 6767	. 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 3209   ⊆ wss 3211  ^◡ccnv 4748  dom cdm 4749   ∘ ccom 4753  Rel wrel 4754   Er wer 6764

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 6767

This theorem is referenced by:  ercl  6778  erref  6787  errn  6789  erssxp  6790  erexb  6792  ereldm  6812  uniqs2  6829  iinerm  6841  th3qlem1  6871  0nnq  7679  nnnq0lem1  7761  prsrlem1  8057  gt0srpr  8063  0nsr  8064  divsfval  13541