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 ) ) ) |