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 = ( 𝐶cat 𝐻 )
rescbas.b 𝐵 = ( Base ‘ 𝐶 )
rescbas.c ( 𝜑𝐶𝑉 )
rescbas.h ( 𝜑𝐻 Fn ( 𝑆 × 𝑆 ) )
rescbas.s ( 𝜑𝑆𝐵 )
Assertion reschomf ( 𝜑𝐻 = ( Homf𝐷 ) )

Proof

Step Hyp Ref Expression
1 rescbas.d 𝐷 = ( 𝐶cat 𝐻 )
2 rescbas.b 𝐵 = ( Base ‘ 𝐶 )
3 rescbas.c ( 𝜑𝐶𝑉 )
4 rescbas.h ( 𝜑𝐻 Fn ( 𝑆 × 𝑆 ) )
5 rescbas.s ( 𝜑𝑆𝐵 )
6 1 2 3 4 5 reschom ( 𝜑𝐻 = ( Hom ‘ 𝐷 ) )
7 1 2 3 4 5 rescbas ( 𝜑𝑆 = ( 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 ( 𝜑𝐻 = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) )
14 eqid ( Homf𝐷 ) = ( Homf𝐷 )
15 eqid ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
16 eqid ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
17 14 15 16 homffval ( Homf𝐷 ) = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
18 13 17 eqtr4di ( 𝜑𝐻 = ( Homf𝐷 ) )