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 |
1 2 3 4 5 6 7
|
funcestrcsetc |
⊢ ( 𝜑 → 𝐹 ( 𝐸 Func 𝑆 ) 𝐺 ) |
9 |
1 2 3 4 5 6 7
|
funcestrcsetclem8 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝑎 𝐺 𝑏 ) : ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ⟶ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ) |
10 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑈 ∈ WUni ) |
11 |
|
eqid |
⊢ ( Hom ‘ 𝑆 ) = ( Hom ‘ 𝑆 ) |
12 |
1 2 3 4 5 6
|
funcestrcsetclem2 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑎 ) ∈ 𝑈 ) |
13 |
12
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑎 ) ∈ 𝑈 ) |
14 |
1 2 3 4 5 6
|
funcestrcsetclem2 |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑏 ) ∈ 𝑈 ) |
15 |
14
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑏 ) ∈ 𝑈 ) |
16 |
2 10 11 13 15
|
elsetchom |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ℎ ∈ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ↔ ℎ : ( 𝐹 ‘ 𝑎 ) ⟶ ( 𝐹 ‘ 𝑏 ) ) ) |
17 |
1 2 3 4 5 6
|
funcestrcsetclem1 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑎 ) = ( Base ‘ 𝑎 ) ) |
18 |
17
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑎 ) = ( Base ‘ 𝑎 ) ) |
19 |
1 2 3 4 5 6
|
funcestrcsetclem1 |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑏 ) = ( Base ‘ 𝑏 ) ) |
20 |
19
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑏 ) = ( Base ‘ 𝑏 ) ) |
21 |
18 20
|
feq23d |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ℎ : ( 𝐹 ‘ 𝑎 ) ⟶ ( 𝐹 ‘ 𝑏 ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
22 |
16 21
|
bitrd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ℎ ∈ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
23 |
|
fvex |
⊢ ( Base ‘ 𝑏 ) ∈ V |
24 |
|
fvex |
⊢ ( Base ‘ 𝑎 ) ∈ V |
25 |
23 24
|
pm3.2i |
⊢ ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) |
26 |
|
elmapg |
⊢ ( ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
27 |
25 26
|
mp1i |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
28 |
27
|
biimpar |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
29 |
|
equequ2 |
⊢ ( 𝑘 = ℎ → ( ℎ = 𝑘 ↔ ℎ = ℎ ) ) |
30 |
29
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ∧ 𝑘 = ℎ ) → ( ℎ = 𝑘 ↔ ℎ = ℎ ) ) |
31 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ = ℎ ) |
32 |
28 30 31
|
rspcedvd |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = 𝑘 ) |
33 |
|
eqid |
⊢ ( Base ‘ 𝑎 ) = ( Base ‘ 𝑎 ) |
34 |
|
eqid |
⊢ ( Base ‘ 𝑏 ) = ( Base ‘ 𝑏 ) |
35 |
1 2 3 4 5 6 7 33 34
|
funcestrcsetclem6 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) → ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) = 𝑘 ) |
36 |
35
|
3expa |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) → ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) = 𝑘 ) |
37 |
36
|
eqeq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) → ( ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ↔ ℎ = 𝑘 ) ) |
38 |
37
|
rexbidva |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ↔ ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = 𝑘 ) ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ( ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ↔ ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = 𝑘 ) ) |
40 |
32 39
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) |
41 |
|
eqid |
⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 ) |
42 |
1 5
|
estrcbas |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐸 ) ) |
43 |
3 42
|
eqtr4id |
⊢ ( 𝜑 → 𝐵 = 𝑈 ) |
44 |
43
|
eleq2d |
⊢ ( 𝜑 → ( 𝑎 ∈ 𝐵 ↔ 𝑎 ∈ 𝑈 ) ) |
45 |
44
|
biimpcd |
⊢ ( 𝑎 ∈ 𝐵 → ( 𝜑 → 𝑎 ∈ 𝑈 ) ) |
46 |
45
|
adantr |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝜑 → 𝑎 ∈ 𝑈 ) ) |
47 |
46
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑎 ∈ 𝑈 ) |
48 |
43
|
eleq2d |
⊢ ( 𝜑 → ( 𝑏 ∈ 𝐵 ↔ 𝑏 ∈ 𝑈 ) ) |
49 |
48
|
biimpcd |
⊢ ( 𝑏 ∈ 𝐵 → ( 𝜑 → 𝑏 ∈ 𝑈 ) ) |
50 |
49
|
adantl |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝜑 → 𝑏 ∈ 𝑈 ) ) |
51 |
50
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑏 ∈ 𝑈 ) |
52 |
1 10 41 47 51 33 34
|
estrchom |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
53 |
52
|
rexeqdv |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ∃ 𝑘 ∈ ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ↔ ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) ) |
54 |
53
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ( ∃ 𝑘 ∈ ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ↔ ∃ 𝑘 ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) ) |
55 |
40 54
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ∃ 𝑘 ∈ ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) |
56 |
55
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) → ∃ 𝑘 ∈ ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) ) |
57 |
22 56
|
sylbid |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ℎ ∈ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) → ∃ 𝑘 ∈ ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) ) |
58 |
57
|
ralrimiv |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ∀ ℎ ∈ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ∃ 𝑘 ∈ ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) |
59 |
|
dffo3 |
⊢ ( ( 𝑎 𝐺 𝑏 ) : ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) –onto→ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ↔ ( ( 𝑎 𝐺 𝑏 ) : ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ⟶ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ∧ ∀ ℎ ∈ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ∃ 𝑘 ∈ ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) ℎ = ( ( 𝑎 𝐺 𝑏 ) ‘ 𝑘 ) ) ) |
60 |
9 58 59
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝑎 𝐺 𝑏 ) : ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) –onto→ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ) |
61 |
60
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 𝐺 𝑏 ) : ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) –onto→ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ) |
62 |
3 11 41
|
isfull2 |
⊢ ( 𝐹 ( 𝐸 Full 𝑆 ) 𝐺 ↔ ( 𝐹 ( 𝐸 Func 𝑆 ) 𝐺 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 𝐺 𝑏 ) : ( 𝑎 ( Hom ‘ 𝐸 ) 𝑏 ) –onto→ ( ( 𝐹 ‘ 𝑎 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑏 ) ) ) ) |
63 |
8 61 62
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 ( 𝐸 Full 𝑆 ) 𝐺 ) |