Theorem fthres2c

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

	Ref	Expression
Hypotheses	fthres2c.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
	fthres2c.e	⊢ 𝐸 = ( 𝐷 ↾s 𝑆 )
	fthres2c.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	fthres2c.r	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
	fthres2c.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 )
Assertion	fthres2c	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Faith 𝐸 ) 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		fthres2c.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
2		fthres2c.e	⊢ 𝐸 = ( 𝐷 ↾s 𝑆 )
3		fthres2c.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
4		fthres2c.r	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
5		fthres2c.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 )
6	1 2 3 4 5	funcres2c	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Func 𝐸 ) 𝐺 ) )
7	6	anbi1d	⊢ ( 𝜑 → ( ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 Fun ◡ ( 𝑥 𝐺 𝑦 ) ) ↔ ( 𝐹 ( 𝐶 Func 𝐸 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 Fun ◡ ( 𝑥 𝐺 𝑦 ) ) ) )
8	1	isfth	⊢ ( 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ↔ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 Fun ◡ ( 𝑥 𝐺 𝑦 ) ) )
9	1	isfth	⊢ ( 𝐹 ( 𝐶 Faith 𝐸 ) 𝐺 ↔ ( 𝐹 ( 𝐶 Func 𝐸 ) 𝐺 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 Fun ◡ ( 𝑥 𝐺 𝑦 ) ) )
10	7 8 9	3bitr4g	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Faith 𝐸 ) 𝐺 ) )