Theorem erdm 6707

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 6697	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1037	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1395   ∪ cun 3196   ⊆ wss 3198  ^◡ccnv 4722  dom cdm 4723   ∘ ccom 4727  Rel wrel 4728   Er wer 6694

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 1004  df-er 6697

This theorem is referenced by:  ercl  6708  erref  6717  errn  6719  erssxp  6720  erexb  6722  ereldm  6742  uniqs2  6759  iinerm  6771  th3qlem1  6801  0nnq  7574  nnnq0lem1  7656  prsrlem1  7952  gt0srpr  7958  0nsr  7959  divsfval  13401