Step | Hyp | Ref
| Expression |
1 | | f1oi 6716 |
. . . 4
⊢ ( I
↾ ((Base‘𝑌)
↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑m (Base‘𝑋)) |
2 | | f1of 6679 |
. . . 4
⊢ (( I
↾ ((Base‘𝑌)
↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑m (Base‘𝑋)) → ( I ↾
((Base‘𝑌)
↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((Base‘𝑌) ↑m
(Base‘𝑋))) |
3 | 1, 2 | mp1i 13 |
. . 3
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ ((Base‘𝑌) ↑m
(Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((Base‘𝑌) ↑m
(Base‘𝑋))) |
4 | | elmapi 8550 |
. . . . 5
⊢ (𝑓 ∈ ((Base‘𝑌) ↑m
(Base‘𝑋)) →
𝑓:(Base‘𝑋)⟶(Base‘𝑌)) |
5 | | fvex 6748 |
. . . . . . . . . 10
⊢
(Base‘𝑌)
∈ V |
6 | | fvex 6748 |
. . . . . . . . . 10
⊢
(Base‘𝑋)
∈ V |
7 | 5, 6 | pm3.2i 474 |
. . . . . . . . 9
⊢
((Base‘𝑌)
∈ V ∧ (Base‘𝑋) ∈ V) |
8 | | elmapg 8541 |
. . . . . . . . . 10
⊢
(((Base‘𝑌)
∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌))) |
9 | 8 | bicomd 226 |
. . . . . . . . 9
⊢
(((Base‘𝑌)
∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))) |
10 | 7, 9 | mp1i 13 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))) |
11 | 10 | biimpa 480 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) |
12 | | funcestrcsetc.e |
. . . . . . . . . . 11
⊢ 𝐸 = (ExtStrCat‘𝑈) |
13 | | funcestrcsetc.s |
. . . . . . . . . . 11
⊢ 𝑆 = (SetCat‘𝑈) |
14 | | funcestrcsetc.b |
. . . . . . . . . . 11
⊢ 𝐵 = (Base‘𝐸) |
15 | | funcestrcsetc.c |
. . . . . . . . . . 11
⊢ 𝐶 = (Base‘𝑆) |
16 | | funcestrcsetc.u |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ∈ WUni) |
17 | | funcestrcsetc.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
18 | 12, 13, 14, 15, 16, 17 | funcestrcsetclem1 17671 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌)) |
19 | 18 | adantrl 716 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌)) |
20 | 12, 13, 14, 15, 16, 17 | funcestrcsetclem1 17671 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
21 | 20 | adantrr 717 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋)) |
22 | 19, 21 | oveq12d 7249 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑌) ↑m (𝐹‘𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋))) |
23 | 22 | adantr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹‘𝑌) ↑m (𝐹‘𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋))) |
24 | 11, 23 | eleqtrrd 2842 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋))) |
25 | 24 | ex 416 |
. . . . 5
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))) |
26 | 4, 25 | syl5 34 |
. . . 4
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))) |
27 | 26 | ssrdv 3921 |
. . 3
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((Base‘𝑌) ↑m (Base‘𝑋)) ⊆ ((𝐹‘𝑌) ↑m (𝐹‘𝑋))) |
28 | 3, 27 | fssd 6581 |
. 2
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ ((Base‘𝑌) ↑m
(Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((𝐹‘𝑌) ↑m (𝐹‘𝑋))) |
29 | | funcestrcsetc.g |
. . . 4
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m
(Base‘𝑥))))) |
30 | | eqid 2738 |
. . . 4
⊢
(Base‘𝑋) =
(Base‘𝑋) |
31 | | eqid 2738 |
. . . 4
⊢
(Base‘𝑌) =
(Base‘𝑌) |
32 | 12, 13, 14, 15, 16, 17, 29, 30, 31 | funcestrcsetclem5 17675 |
. . 3
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑m
(Base‘𝑋)))) |
33 | 16 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑈 ∈ WUni) |
34 | | eqid 2738 |
. . . 4
⊢ (Hom
‘𝐸) = (Hom
‘𝐸) |
35 | 12, 16 | estrcbas 17656 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 = (Base‘𝐸)) |
36 | 14, 35 | eqtr4id 2798 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = 𝑈) |
37 | 36 | eleq2d 2824 |
. . . . . . 7
⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈)) |
38 | 37 | biimpcd 252 |
. . . . . 6
⊢ (𝑋 ∈ 𝐵 → (𝜑 → 𝑋 ∈ 𝑈)) |
39 | 38 | adantr 484 |
. . . . 5
⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝜑 → 𝑋 ∈ 𝑈)) |
40 | 39 | impcom 411 |
. . . 4
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝑈) |
41 | 36 | eleq2d 2824 |
. . . . . . 7
⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ 𝑈)) |
42 | 41 | biimpd 232 |
. . . . . 6
⊢ (𝜑 → (𝑌 ∈ 𝐵 → 𝑌 ∈ 𝑈)) |
43 | 42 | adantld 494 |
. . . . 5
⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝑈)) |
44 | 43 | imp 410 |
. . . 4
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝑈) |
45 | 12, 33, 34, 40, 44, 30, 31 | estrchom 17658 |
. . 3
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋))) |
46 | | eqid 2738 |
. . . 4
⊢ (Hom
‘𝑆) = (Hom
‘𝑆) |
47 | 12, 13, 14, 15, 16, 17 | funcestrcsetclem2 17672 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈) |
48 | 47 | adantrr 717 |
. . . 4
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈) |
49 | 12, 13, 14, 15, 16, 17 | funcestrcsetclem2 17672 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈) |
50 | 49 | adantrl 716 |
. . . 4
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈) |
51 | 13, 33, 46, 48, 50 | setchom 17610 |
. . 3
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) = ((𝐹‘𝑌) ↑m (𝐹‘𝑋))) |
52 | 32, 45, 51 | feq123d 6552 |
. 2
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) ↔ ( I ↾ ((Base‘𝑌) ↑m
(Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((𝐹‘𝑌) ↑m (𝐹‘𝑋)))) |
53 | 28, 52 | mpbird 260 |
1
⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌))) |