Theorem qliftlem

Description: Lemma for theorems about a function lift. (Contributed by Mario Carneiro, 23-Dec-2016) (Revised by AV, 3-Aug-2024)

	Ref	Expression
Hypotheses	qlift.1	⊢ 𝐹 = ran ( 𝑥 ∈ 𝑋 ↦ ⟨ [ 𝑥 ] 𝑅 , 𝐴 ⟩ )
	qlift.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ 𝑌 )
	qlift.3	⊢ ( 𝜑 → 𝑅 Er 𝑋 )
	qlift.4	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
Assertion	qliftlem	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → [ 𝑥 ] 𝑅 ∈ ( 𝑋 / 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		qlift.1	⊢ 𝐹 = ran ( 𝑥 ∈ 𝑋 ↦ ⟨ [ 𝑥 ] 𝑅 , 𝐴 ⟩ )
2		qlift.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ 𝑌 )
3		qlift.3	⊢ ( 𝜑 → 𝑅 Er 𝑋 )
4		qlift.4	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
5		erex	⊢ ( 𝑅 Er 𝑋 → ( 𝑋 ∈ 𝑉 → 𝑅 ∈ V ) )
6	3 4 5	sylc	⊢ ( 𝜑 → 𝑅 ∈ V )
7		ecelqsw	⊢ ( ( 𝑅 ∈ V ∧ 𝑥 ∈ 𝑋 ) → [ 𝑥 ] 𝑅 ∈ ( 𝑋 / 𝑅 ) )
8	6 7	sylan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → [ 𝑥 ] 𝑅 ∈ ( 𝑋 / 𝑅 ) )