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Metamath Proof Explorer

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 ) )