Theorem qliftval

Description: The value of the function F . (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 )
	qliftval.4	⊢ ( 𝑥 = 𝐶 → 𝐴 = 𝐵 )
	qliftval.6	⊢ ( 𝜑 → Fun 𝐹 )
Assertion	qliftval	⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ [ 𝐶 ] 𝑅 ) = 𝐵 )

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		qliftval.4	⊢ ( 𝑥 = 𝐶 → 𝐴 = 𝐵 )
6		qliftval.6	⊢ ( 𝜑 → Fun 𝐹 )
7	1 2 3 4	qliftlem	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → [ 𝑥 ] 𝑅 ∈ ( 𝑋 / 𝑅 ) )
8		eceq1	⊢ ( 𝑥 = 𝐶 → [ 𝑥 ] 𝑅 = [ 𝐶 ] 𝑅 )
9	1 7 2 8 5 6	fliftval	⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ [ 𝐶 ] 𝑅 ) = 𝐵 )