Theorem erdm 6439

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 6429	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp2bi 997	1 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1331   ∪ cun 3069   ⊆ wss 3071  ^◡ccnv 4538  dom cdm 4539   ∘ ccom 4543  Rel wrel 4544   Er wer 6426

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 964  df-er 6429

This theorem is referenced by:  ercl  6440  erref  6449  errn  6451  erssxp  6452  erexb  6454  ereldm  6472  uniqs2  6489  iinerm  6501  th3qlem1  6531  0nnq  7172  nnnq0lem1  7254  prsrlem1  7550  gt0srpr  7556  0nsr  7557