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Metamath Proof Explorer

Theorem funcestrcsetclem7

Description: Lemma 7 for funcestrcsetc . (Contributed by AV, 23-Mar-2020)

Ref Expression
Hypotheses funcestrcsetc.e ⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
funcestrcsetc.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
funcestrcsetc.b ⊢ 𝐵 = ( Base ‘ 𝐸 )
funcestrcsetc.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
funcestrcsetc.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
funcestrcsetc.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
funcestrcsetc.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) )
Assertion funcestrcsetclem7 ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝐸 ) ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 funcestrcsetc.e ⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
2 funcestrcsetc.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3 funcestrcsetc.b ⊢ 𝐵 = ( Base ‘ 𝐸 )
4 funcestrcsetc.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
5 funcestrcsetc.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
6 funcestrcsetc.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 funcestrcsetc.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) )
8 eqid ⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 )
9 1 2 3 4 5 6 7 8 8 funcestrcsetclem5 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑋 ) = ( I ↾ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) ) )
10 9 anabsan2 ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 𝐺 𝑋 ) = ( I ↾ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) ) )
11 eqid ⊢ ( Id ‘ 𝐸 ) = ( Id ‘ 𝐸 )
12 5 adantr ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑈 ∈ WUni )
13 1 5 estrcbas ⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐸 ) )
14 3 13 eqtr4id ⊢ ( 𝜑 → 𝐵 = 𝑈 )
15 14 eleq2d ⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈 ) )
16 15 biimpa ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝑈 )
17 1 11 12 16 estrcid ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( Id ‘ 𝐸 ) ‘ 𝑋 ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
18 10 17 fveq12d ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝐸 ) ‘ 𝑋 ) ) = ( ( I ↾ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) ) ‘ ( I ↾ ( Base ‘ 𝑋 ) ) ) )
19 fvex ⊢ ( Base ‘ 𝑋 ) ∈ V
20 19 19 pm3.2i ⊢ ( ( Base ‘ 𝑋 ) ∈ V ∧ ( Base ‘ 𝑋 ) ∈ V )
21 20 a1i ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( Base ‘ 𝑋 ) ∈ V ∧ ( Base ‘ 𝑋 ) ∈ V ) )
22 f1oi ⊢ ( I ↾ ( Base ‘ 𝑋 ) ) : ( Base ‘ 𝑋 ) –1-1-onto→ ( Base ‘ 𝑋 )
23 f1of ⊢ ( ( I ↾ ( Base ‘ 𝑋 ) ) : ( Base ‘ 𝑋 ) –1-1-onto→ ( Base ‘ 𝑋 ) → ( I ↾ ( Base ‘ 𝑋 ) ) : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑋 ) )
24 22 23 ax-mp ⊢ ( I ↾ ( Base ‘ 𝑋 ) ) : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑋 )
25 elmapg ⊢ ( ( ( Base ‘ 𝑋 ) ∈ V ∧ ( Base ‘ 𝑋 ) ∈ V ) → ( ( I ↾ ( Base ‘ 𝑋 ) ) ∈ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) ↔ ( I ↾ ( Base ‘ 𝑋 ) ) : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑋 ) ) )
26 24 25 mpbiri ⊢ ( ( ( Base ‘ 𝑋 ) ∈ V ∧ ( Base ‘ 𝑋 ) ∈ V ) → ( I ↾ ( Base ‘ 𝑋 ) ) ∈ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) )
27 fvresi ⊢ ( ( I ↾ ( Base ‘ 𝑋 ) ) ∈ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) → ( ( I ↾ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) ) ‘ ( I ↾ ( Base ‘ 𝑋 ) ) ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
28 21 26 27 3syl ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( I ↾ ( ( Base ‘ 𝑋 ) ↑m ( Base ‘ 𝑋 ) ) ) ‘ ( I ↾ ( Base ‘ 𝑋 ) ) ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
29 1 2 3 4 5 6 funcestrcsetclem1 ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )
30 29 fveq2d ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( Base ‘ 𝑋 ) ) )
31 eqid ⊢ ( Id ‘ 𝑆 ) = ( Id ‘ 𝑆 )
32 1 3 5 estrcbasbas ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( Base ‘ 𝑋 ) ∈ 𝑈 )
33 2 31 12 32 setcid ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( Id ‘ 𝑆 ) ‘ ( Base ‘ 𝑋 ) ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
34 30 33 eqtr2d ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( I ↾ ( Base ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) )
35 18 28 34 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝐸 ) ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) )