Theorem refrelcosslem

Description: Lemma for the left side of the refrelcoss3 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019)

	Ref	Expression
Assertion	refrelcosslem	⊢ ∀ 𝑥 ∈ dom ≀ 𝑅 𝑥 ≀ 𝑅 𝑥

Proof

Step	Hyp	Ref	Expression
1		ralel	⊢ ∀ 𝑥 ∈ dom ≀ 𝑅 𝑥 ∈ dom ≀ 𝑅
2		eldmcoss2	⊢ ( 𝑥 ∈ V → ( 𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅 𝑥 ) )
3	2	elv	⊢ ( 𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅 𝑥 )
4	3	ralbii	⊢ ( ∀ 𝑥 ∈ dom ≀ 𝑅 𝑥 ∈ dom ≀ 𝑅 ↔ ∀ 𝑥 ∈ dom ≀ 𝑅 𝑥 ≀ 𝑅 𝑥 )
5	1 4	mpbi	⊢ ∀ 𝑥 ∈ dom ≀ 𝑅 𝑥 ≀ 𝑅 𝑥