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Metamath Proof Explorer

Theorem elestrchom

Description: A morphism between extensible structures is a function 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 elestrchom ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )

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 4 5 6 7 estrchom ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝐵 ↑m 𝐴 ) )
9 8 eleq2d ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) ) )
10 7 fvexi ⊢ 𝐵 ∈ V
11 10 a1i ⊢ ( 𝜑 → 𝐵 ∈ V )
12 6 fvexi ⊢ 𝐴 ∈ V
13 12 a1i ⊢ ( 𝜑 → 𝐴 ∈ V )
14 11 13 elmapd ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )
15 9 14 bitrd ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )