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Metamath Proof Explorer

Theorem estrchomfval

Description: Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020)

Ref Expression
Hypotheses estrcbas.c ⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
estrcbas.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
estrchomfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion estrchomfval ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )

Proof

Step Hyp Ref Expression
1 estrcbas.c ⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
2 estrcbas.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3 estrchomfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
4 eqidd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
5 eqidd ⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) = ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )
6 1 2 4 5 estrcval ⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } )
7 catstr ⊢ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } Struct ⟨ 1 , 1 5 ⟩
8 homid ⊢ Hom = Slot ( Hom ‘ ndx )
9 snsstp2 ⊢ { ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ }
10 mpoexga ⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉 ) → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ∈ V )
11 2 2 10 syl2anc ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ∈ V )
12 6 7 8 9 11 3 strfv3 ⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )