Theorem erdm 6807

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 6797	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1044	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1402   ∪ cun 3218   ⊆ wss 3220  ^◡ccnv 4768  dom cdm 4769   ∘ ccom 4773  Rel wrel 4774   Er wer 6794

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 1011  df-er 6797

This theorem is referenced by:  ercl  6808  erref  6817  errn  6819  erssxp  6820  erexb  6822  ereldm  6842  uniqs2  6859  iinerm  6871  th3qlem1  6901  0nnq  7721  nnnq0lem1  7803  prsrlem1  8099  gt0srpr  8105  0nsr  8106  divsfval  13626