Theorem erdm 6547

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 6537	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 1013	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1353   ∪ cun 3129   ⊆ wss 3131  ^◡ccnv 4627  dom cdm 4628   ∘ ccom 4632  Rel wrel 4633   Er wer 6534

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 980  df-er 6537

This theorem is referenced by:  ercl  6548  erref  6557  errn  6559  erssxp  6560  erexb  6562  ereldm  6580  uniqs2  6597  iinerm  6609  th3qlem1  6639  0nnq  7365  nnnq0lem1  7447  prsrlem1  7743  gt0srpr  7749  0nsr  7750