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Metamath Proof Explorer

Theorem estrchom

Description: The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020)

Ref Expression
Hypotheses estrcbas.c ⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
estrcbas.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
estrchomfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
estrchom.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑈 )
estrchom.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑈 )
estrchom.a ⊢ 𝐴 = ( Base ‘ 𝑋 )
estrchom.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
Assertion estrchom ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝐵 ↑m 𝐴 ) )

Proof

Step Hyp Ref Expression
1 estrcbas.c ⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
2 estrcbas.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3 estrchomfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
4 estrchom.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑈 )
5 estrchom.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑈 )
6 estrchom.a ⊢ 𝐴 = ( Base ‘ 𝑋 )
7 estrchom.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
8 1 2 3 estrchomfval ⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
9 fveq2 ⊢ ( 𝑦 = 𝑌 → ( Base ‘ 𝑦 ) = ( Base ‘ 𝑌 ) )
10 fveq2 ⊢ ( 𝑥 = 𝑋 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
11 9 10 oveqan12rd ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) )
12 7 6 oveq12i ⊢ ( 𝐵 ↑m 𝐴 ) = ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) )
13 11 12 eqtr4di ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( 𝐵 ↑m 𝐴 ) )
14 13 adantl ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( 𝐵 ↑m 𝐴 ) )
15 ovexd ⊢ ( 𝜑 → ( 𝐵 ↑m 𝐴 ) ∈ V )
16 8 14 4 5 15 ovmpod ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝐵 ↑m 𝐴 ) )