Metamath Proof Explorer


Theorem estrchomfn

Description: The Hom-set operation in the category of extensible structures (in a universe) is a function. (Contributed by AV, 8-Mar-2020)

Ref Expression
Hypotheses estrchomfn.c 𝐶 = ( ExtStrCat ‘ 𝑈 )
estrchomfn.u ( 𝜑𝑈𝑉 )
estrchomfn.h 𝐻 = ( Hom ‘ 𝐶 )
Assertion estrchomfn ( 𝜑𝐻 Fn ( 𝑈 × 𝑈 ) )

Proof

Step Hyp Ref Expression
1 estrchomfn.c 𝐶 = ( ExtStrCat ‘ 𝑈 )
2 estrchomfn.u ( 𝜑𝑈𝑉 )
3 estrchomfn.h 𝐻 = ( Hom ‘ 𝐶 )
4 eqid ( 𝑥𝑈 , 𝑦𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥𝑈 , 𝑦𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) )
5 ovex ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V
6 4 5 fnmpoi ( 𝑥𝑈 , 𝑦𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑈 × 𝑈 )
7 1 2 3 estrchomfval ( 𝜑𝐻 = ( 𝑥𝑈 , 𝑦𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
8 7 fneq1d ( 𝜑 → ( 𝐻 Fn ( 𝑈 × 𝑈 ) ↔ ( 𝑥𝑈 , 𝑦𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑈 × 𝑈 ) ) )
9 6 8 mpbiri ( 𝜑𝐻 Fn ( 𝑈 × 𝑈 ) )