Theorem ffthres2c

Description: Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-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	ffthres2c	⊢ ( 𝜑 → ( 𝐹 ( ( 𝐶 Full 𝐷 ) ∩ ( 𝐶 Faith 𝐷 ) ) 𝐺 ↔ 𝐹 ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 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	fullres2c	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Full 𝐸 ) 𝐺 ) )
7	1 2 3 4 5	fthres2c	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Faith 𝐸 ) 𝐺 ) )
8	6 7	anbi12d	⊢ ( 𝜑 → ( ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ∧ 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ) ↔ ( 𝐹 ( 𝐶 Full 𝐸 ) 𝐺 ∧ 𝐹 ( 𝐶 Faith 𝐸 ) 𝐺 ) ) )
9		brin	⊢ ( 𝐹 ( ( 𝐶 Full 𝐷 ) ∩ ( 𝐶 Faith 𝐷 ) ) 𝐺 ↔ ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ∧ 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ) )
10		brin	⊢ ( 𝐹 ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) 𝐺 ↔ ( 𝐹 ( 𝐶 Full 𝐸 ) 𝐺 ∧ 𝐹 ( 𝐶 Faith 𝐸 ) 𝐺 ) )
11	8 9 10	3bitr4g	⊢ ( 𝜑 → ( 𝐹 ( ( 𝐶 Full 𝐷 ) ∩ ( 𝐶 Faith 𝐷 ) ) 𝐺 ↔ 𝐹 ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) 𝐺 ) )