Theorem qliftf

Description: The domain and range 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 )
Assertion	qliftf	⊢ ( 𝜑 → ( Fun 𝐹 ↔ 𝐹 : ( 𝑋 / 𝑅 ) ⟶ 𝑌 ) )

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	1 2 3 4	qliftlem	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → [ 𝑥 ] 𝑅 ∈ ( 𝑋 / 𝑅 ) )
6	1 5 2	fliftf	⊢ ( 𝜑 → ( Fun 𝐹 ↔ 𝐹 : ran ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] 𝑅 ) ⟶ 𝑌 ) )
7		df-qs	⊢ ( 𝑋 / 𝑅 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝑋 𝑦 = [ 𝑥 ] 𝑅 }
8		eqid	⊢ ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] 𝑅 ) = ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] 𝑅 )
9	8	rnmpt	⊢ ran ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] 𝑅 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝑋 𝑦 = [ 𝑥 ] 𝑅 }
10	7 9	eqtr4i	⊢ ( 𝑋 / 𝑅 ) = ran ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] 𝑅 )
11	10	a1i	⊢ ( 𝜑 → ( 𝑋 / 𝑅 ) = ran ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] 𝑅 ) )
12	11	feq2d	⊢ ( 𝜑 → ( 𝐹 : ( 𝑋 / 𝑅 ) ⟶ 𝑌 ↔ 𝐹 : ran ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] 𝑅 ) ⟶ 𝑌 ) )
13	6 12	bitr4d	⊢ ( 𝜑 → ( Fun 𝐹 ↔ 𝐹 : ( 𝑋 / 𝑅 ) ⟶ 𝑌 ) )