Theorem erdm 6544

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 6534	. 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 3127   ⊆ wss 3129  ^◡ccnv 4625  dom cdm 4626   ∘ ccom 4630  Rel wrel 4631   Er wer 6531

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 6534

This theorem is referenced by:  ercl  6545  erref  6554  errn  6556  erssxp  6557  erexb  6559  ereldm  6577  uniqs2  6594  iinerm  6606  th3qlem1  6636  0nnq  7362  nnnq0lem1  7444  prsrlem1  7740  gt0srpr  7746  0nsr  7747