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Metamath Proof Explorer

Theorem estrchomfeqhom

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

Ref Expression
Hypotheses estrchomfn.c ⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
estrchomfn.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
estrchomfn.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion estrchomfeqhom ( 𝜑 → ( Homf ‘ 𝐶 ) = 𝐻 )

Proof

Step Hyp Ref Expression
1 estrchomfn.c ⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
2 estrchomfn.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3 estrchomfn.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
4 1 2 3 estrchomfn ⊢ ( 𝜑 → 𝐻 Fn ( 𝑈 × 𝑈 ) )
5 1 2 estrcbas ⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐶 ) )
6 5 eqcomd ⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = 𝑈 )
7 6 sqxpeqd ⊢ ( 𝜑 → ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) = ( 𝑈 × 𝑈 ) )
8 7 fneq2d ⊢ ( 𝜑 → ( 𝐻 Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ↔ 𝐻 Fn ( 𝑈 × 𝑈 ) ) )
9 4 8 mpbird ⊢ ( 𝜑 → 𝐻 Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) )
10 eqid ⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐶 )
11 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
12 10 11 3 fnhomeqhomf ⊢ ( 𝐻 Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = 𝐻 )
13 9 12 syl ⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = 𝐻 )