Theorem erdm 6611

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 6601	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1015	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1364   ∪ cun 3155   ⊆ wss 3157  ^◡ccnv 4663  dom cdm 4664   ∘ ccom 4668  Rel wrel 4669   Er wer 6598

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 982  df-er 6601

This theorem is referenced by:  ercl  6612  erref  6621  errn  6623  erssxp  6624  erexb  6626  ereldm  6646  uniqs2  6663  iinerm  6675  th3qlem1  6705  0nnq  7448  nnnq0lem1  7530  prsrlem1  7826  gt0srpr  7832  0nsr  7833  divsfval  13030