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Metamath Proof Explorer

Theorem reschomf

Description: Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses rescbas.d ⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 )
rescbas.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
rescbas.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
rescbas.h ⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
rescbas.s ⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
Assertion reschomf ( 𝜑 → 𝐻 = ( Homf ‘ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 rescbas.d ⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 )
2 rescbas.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 rescbas.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4 rescbas.h ⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
5 rescbas.s ⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
6 1 2 3 4 5 reschom ⊢ ( 𝜑 → 𝐻 = ( Hom ‘ 𝐷 ) )
7 1 2 3 4 5 rescbas ⊢ ( 𝜑 → 𝑆 = ( Base ‘ 𝐷 ) )
8 7 sqxpeqd ⊢ ( 𝜑 → ( 𝑆 × 𝑆 ) = ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) )
9 6 8 fneq12d ⊢ ( 𝜑 → ( 𝐻 Fn ( 𝑆 × 𝑆 ) ↔ ( Hom ‘ 𝐷 ) Fn ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) ) )
10 4 9 mpbid ⊢ ( 𝜑 → ( Hom ‘ 𝐷 ) Fn ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) )
11 fnov ⊢ ( ( Hom ‘ 𝐷 ) Fn ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) ↔ ( Hom ‘ 𝐷 ) = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) )
12 10 11 sylib ⊢ ( 𝜑 → ( Hom ‘ 𝐷 ) = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) )
13 6 12 eqtrd ⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) )
14 eqid ⊢ ( Homf ‘ 𝐷 ) = ( Homf ‘ 𝐷 )
15 eqid ⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
16 eqid ⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
17 14 15 16 homffval ⊢ ( Homf ‘ 𝐷 ) = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
18 13 17 eqtr4di ⊢ ( 𝜑 → 𝐻 = ( Homf ‘ 𝐷 ) )