Theorem erdm 6755

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 6745	. 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 3199   ⊆ wss 3201  ^◡ccnv 4730  dom cdm 4731   ∘ ccom 4735  Rel wrel 4736   Er wer 6742

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 6745

This theorem is referenced by:  ercl  6756  erref  6765  errn  6767  erssxp  6768  erexb  6770  ereldm  6790  uniqs2  6807  iinerm  6819  th3qlem1  6849  0nnq  7627  nnnq0lem1  7709  prsrlem1  8005  gt0srpr  8011  0nsr  8012  divsfval  13474