Theorem erdm 6393

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 6383	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 980	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1314   ∪ cun 3035   ⊆ wss 3037  ^◡ccnv 4498  dom cdm 4499   ∘ ccom 4503  Rel wrel 4504   Er wer 6380

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

This theorem depends on definitions:  df-bi 116  df-3an 947  df-er 6383

This theorem is referenced by:  ercl  6394  erref  6403  errn  6405  erssxp  6406  erexb  6408  ereldm  6426  uniqs2  6443  iinerm  6455  th3qlem1  6485  0nnq  7120  nnnq0lem1  7202  prsrlem1  7485  gt0srpr  7491  0nsr  7492