Theorem erdm 6653

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 6643	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1016	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1373   ∪ cun 3172   ⊆ wss 3174  ^◡ccnv 4692  dom cdm 4693   ∘ ccom 4697  Rel wrel 4698   Er wer 6640

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 983  df-er 6643

This theorem is referenced by:  ercl  6654  erref  6663  errn  6665  erssxp  6666  erexb  6668  ereldm  6688  uniqs2  6705  iinerm  6717  th3qlem1  6747  0nnq  7512  nnnq0lem1  7594  prsrlem1  7890  gt0srpr  7896  0nsr  7897  divsfval  13275