Theorem erdm 6822

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 6812	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1044	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∪ cun 3218   ⊆ wss 3220  ^◡ccnv 4775  dom cdm 4776   ∘ ccom 4780  Rel wrel 4781   Er wer 6809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-3an 1011  df-er 6812

This theorem is used by:  ercl  6823  erref  6832  errn  6834  erssxp  6835  erexb  6837  ereldm  6857  uniqs2  6874  iinerm  6886  th3qlem1  6916  0nnq  7737  nnnq0lem1  7819  prsrlem1  8115  gt0srpr  8121  0nsr  8122  divsfval  13707