Theorem rncossdmcoss

Description: The range of cosets is the domain of them (this should be rncoss but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019)

	Ref	Expression
Assertion	rncossdmcoss	⊢ ran ≀ 𝑅 = dom ≀ 𝑅

Proof

Step	Hyp	Ref	Expression
1		brcosscnvcoss	⊢ ( ( 𝑦 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑦 ≀ 𝑅 𝑥 ↔ 𝑥 ≀ 𝑅 𝑦 ) )
2	1	el2v	⊢ ( 𝑦 ≀ 𝑅 𝑥 ↔ 𝑥 ≀ 𝑅 𝑦 )
3	2	exbii	⊢ ( ∃ 𝑦 𝑦 ≀ 𝑅 𝑥 ↔ ∃ 𝑦 𝑥 ≀ 𝑅 𝑦 )
4	3	abbii	⊢ { 𝑥 ∣ ∃ 𝑦 𝑦 ≀ 𝑅 𝑥 } = { 𝑥 ∣ ∃ 𝑦 𝑥 ≀ 𝑅 𝑦 }
5		dfrn2	⊢ ran ≀ 𝑅 = { 𝑥 ∣ ∃ 𝑦 𝑦 ≀ 𝑅 𝑥 }
6		df-dm	⊢ dom ≀ 𝑅 = { 𝑥 ∣ ∃ 𝑦 𝑥 ≀ 𝑅 𝑦 }
7	4 5 6	3eqtr4i	⊢ ran ≀ 𝑅 = dom ≀ 𝑅