Theorem erdm 6511

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 6501	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1003	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1343   ∪ cun 3114   ⊆ wss 3116  ^◡ccnv 4603  dom cdm 4604   ∘ ccom 4608  Rel wrel 4609   Er wer 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-3an 970  df-er 6501

This theorem is referenced by:  ercl  6512  erref  6521  errn  6523  erssxp  6524  erexb  6526  ereldm  6544  uniqs2  6561  iinerm  6573  th3qlem1  6603  0nnq  7305  nnnq0lem1  7387  prsrlem1  7683  gt0srpr  7689  0nsr  7690