This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )