Theorem fullres2c

Description: Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017)

	Ref	Expression
Hypotheses	ffthres2c.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
	ffthres2c.e	⊢ 𝐸 = ( 𝐷 ↾s 𝑆 )
	ffthres2c.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	ffthres2c.r	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
	ffthres2c.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 )
Assertion	fullres2c	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Full 𝐸 ) 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		ffthres2c.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
2		ffthres2c.e	⊢ 𝐸 = ( 𝐷 ↾s 𝑆 )
3		ffthres2c.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
4		ffthres2c.r	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
5		ffthres2c.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 )
6	1 2 3 4 5	funcres2c	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Func 𝐸 ) 𝐺 ) )
7		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
8	2 7	resshom	⊢ ( 𝑆 ∈ 𝑉 → ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐸 ) )
9	4 8	syl	⊢ ( 𝜑 → ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐸 ) )
10	9	oveqd	⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) )
11	10	eqeq2d	⊢ ( 𝜑 → ( ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) ) )
12	11	2ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) ) )
13	6 12	anbi12d	⊢ ( 𝜑 → ( ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( 𝐹 ( 𝐶 Func 𝐸 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) ) ) )
14	1 7	isfull	⊢ ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ↔ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) )
15		eqid	⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 )
16	1 15	isfull	⊢ ( 𝐹 ( 𝐶 Full 𝐸 ) 𝐺 ↔ ( 𝐹 ( 𝐶 Func 𝐸 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ran ( 𝑥 𝐺 𝑦 ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) ) )
17	13 14 16	3bitr4g	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Full 𝐸 ) 𝐺 ) )