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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 ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
estrchomfn.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion estrchomfn ( 𝜑 → 𝐻 Fn ( 𝑈 × 𝑈 ) )

Proof

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