Proof of Theorem rescabs
Step | Hyp | Ref
| Expression |
1 | | eqid 2737 |
. . . 4
⊢ (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx),
𝐻〉)
↾cat 𝐽) =
(((𝐶 ↾s
𝑆) sSet 〈(Hom
‘ndx), 𝐻〉)
↾cat 𝐽) |
2 | | ovexd 7248 |
. . . 4
⊢ (𝜑 → ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ∈
V) |
3 | | rescabs.s |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ 𝑊) |
4 | | rescabs.t |
. . . . 5
⊢ (𝜑 → 𝑇 ⊆ 𝑆) |
5 | 3, 4 | ssexd 5217 |
. . . 4
⊢ (𝜑 → 𝑇 ∈ V) |
6 | | rescabs.j |
. . . 4
⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) |
7 | 1, 2, 5, 6 | rescval2 17333 |
. . 3
⊢ (𝜑 → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾cat
𝐽) = ((((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉)) |
8 | | simpr 488 |
. . . . . . 7
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) |
9 | | ovexd 7248 |
. . . . . . 7
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ∈
V) |
10 | 5 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V) |
11 | | eqid 2737 |
. . . . . . . 8
⊢ (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx),
𝐻〉)
↾s 𝑇) =
(((𝐶 ↾s
𝑆) sSet 〈(Hom
‘ndx), 𝐻〉)
↾s 𝑇) |
12 | | baseid 16763 |
. . . . . . . . 9
⊢ Base =
Slot (Base‘ndx) |
13 | | 1re 10833 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ |
14 | | 1nn 11841 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ |
15 | | 4nn0 12109 |
. . . . . . . . . . . 12
⊢ 4 ∈
ℕ0 |
16 | | 1nn0 12106 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ0 |
17 | | 1lt10 12432 |
. . . . . . . . . . . 12
⊢ 1 <
;10 |
18 | 14, 15, 16, 17 | declti 12331 |
. . . . . . . . . . 11
⊢ 1 <
;14 |
19 | 13, 18 | ltneii 10945 |
. . . . . . . . . 10
⊢ 1 ≠
;14 |
20 | | basendx 16769 |
. . . . . . . . . . 11
⊢
(Base‘ndx) = 1 |
21 | | homndx 16918 |
. . . . . . . . . . 11
⊢ (Hom
‘ndx) = ;14 |
22 | 20, 21 | neeq12i 3007 |
. . . . . . . . . 10
⊢
((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
23 | 19, 22 | mpbir 234 |
. . . . . . . . 9
⊢
(Base‘ndx) ≠ (Hom ‘ndx) |
24 | 12, 23 | setsnid 16759 |
. . . . . . . 8
⊢
(Base‘(𝐶
↾s 𝑆)) =
(Base‘((𝐶
↾s 𝑆) sSet
〈(Hom ‘ndx), 𝐻〉)) |
25 | 11, 24 | ressid2 16788 |
. . . . . . 7
⊢
(((Base‘(𝐶
↾s 𝑆))
⊆ 𝑇 ∧ ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx),
𝐻〉) ∈ V ∧
𝑇 ∈ V) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx),
𝐻〉)
↾s 𝑇) =
((𝐶 ↾s
𝑆) sSet 〈(Hom
‘ndx), 𝐻〉)) |
26 | 8, 9, 10, 25 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
27 | 26 | oveq1d 7228 |
. . . . 5
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉) =
(((𝐶 ↾s
𝑆) sSet 〈(Hom
‘ndx), 𝐻〉) sSet
〈(Hom ‘ndx), 𝐽〉)) |
28 | | ovex 7246 |
. . . . . 6
⊢ (𝐶 ↾s 𝑆) ∈ V |
29 | 5, 5 | xpexd 7536 |
. . . . . . . 8
⊢ (𝜑 → (𝑇 × 𝑇) ∈ V) |
30 | | fnex 7033 |
. . . . . . . 8
⊢ ((𝐽 Fn (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ∈ V) → 𝐽 ∈ V) |
31 | 6, 29, 30 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ V) |
32 | 31 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V) |
33 | | setsabs 16732 |
. . . . . 6
⊢ (((𝐶 ↾s 𝑆) ∈ V ∧ 𝐽 ∈ V) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx),
𝐻〉) sSet 〈(Hom
‘ndx), 𝐽〉) =
((𝐶 ↾s
𝑆) sSet 〈(Hom
‘ndx), 𝐽〉)) |
34 | 28, 32, 33 | sylancr 590 |
. . . . 5
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) sSet 〈(Hom
‘ndx), 𝐽〉) =
((𝐶 ↾s
𝑆) sSet 〈(Hom
‘ndx), 𝐽〉)) |
35 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆) |
36 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝐶) =
(Base‘𝐶) |
37 | 35, 36 | ressbas 16790 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ 𝑊 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑆))) |
38 | 3, 37 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑆))) |
39 | 38 | sseq1d 3932 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇 ↔ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇)) |
40 | 39 | biimpar 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇) |
41 | | inss2 4144 |
. . . . . . . . . . 11
⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶) |
42 | 41 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)) |
43 | 40, 42 | ssind 4147 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (𝑇 ∩ (Base‘𝐶))) |
44 | 4 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆) |
45 | 44 | ssrind 4150 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶))) |
46 | 43, 45 | eqssd 3918 |
. . . . . . . 8
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) = (𝑇 ∩ (Base‘𝐶))) |
47 | 46 | oveq2d 7229 |
. . . . . . 7
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶)))) |
48 | 3 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑆 ∈ 𝑊) |
49 | 36 | ressinbas 16797 |
. . . . . . . 8
⊢ (𝑆 ∈ 𝑊 → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))) |
50 | 48, 49 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))) |
51 | 36 | ressinbas 16797 |
. . . . . . . 8
⊢ (𝑇 ∈ V → (𝐶 ↾s 𝑇) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶)))) |
52 | 10, 51 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑇) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶)))) |
53 | 47, 50, 52 | 3eqtr4d 2787 |
. . . . . 6
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑇)) |
54 | 53 | oveq1d 7228 |
. . . . 5
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐽〉) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
55 | 27, 34, 54 | 3eqtrd 2781 |
. . . 4
⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉) =
((𝐶 ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉)) |
56 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) |
57 | | ovexd 7248 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ∈
V) |
58 | 5 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V) |
59 | 11, 24 | ressval2 16789 |
. . . . . . . 8
⊢ ((¬
(Base‘(𝐶
↾s 𝑆))
⊆ 𝑇 ∧ ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx),
𝐻〉) ∈ V ∧
𝑇 ∈ V) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx),
𝐻〉)
↾s 𝑇) =
(((𝐶 ↾s
𝑆) sSet 〈(Hom
‘ndx), 𝐻〉) sSet
〈(Base‘ndx), (𝑇
∩ (Base‘(𝐶
↾s 𝑆)))〉)) |
60 | 56, 57, 58, 59 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) = (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) sSet
〈(Base‘ndx), (𝑇
∩ (Base‘(𝐶
↾s 𝑆)))〉)) |
61 | | ovexd 7248 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) ∈ V) |
62 | 23 | necomi 2995 |
. . . . . . . . 9
⊢ (Hom
‘ndx) ≠ (Base‘ndx) |
63 | 62 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (Hom ‘ndx) ≠
(Base‘ndx)) |
64 | | rescabs.h |
. . . . . . . . . 10
⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
65 | 3, 3 | xpexd 7536 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 × 𝑆) ∈ V) |
66 | | fnex 7033 |
. . . . . . . . . 10
⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) |
67 | 64, 65, 66 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → 𝐻 ∈ V) |
68 | 67 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐻 ∈ V) |
69 | | fvex 6730 |
. . . . . . . . . 10
⊢
(Base‘(𝐶
↾s 𝑆))
∈ V |
70 | 69 | inex2 5211 |
. . . . . . . . 9
⊢ (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V |
71 | 70 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V) |
72 | | fvex 6730 |
. . . . . . . . 9
⊢ (Hom
‘ndx) ∈ V |
73 | | fvex 6730 |
. . . . . . . . 9
⊢
(Base‘ndx) ∈ V |
74 | 72, 73 | setscom 16733 |
. . . . . . . 8
⊢ ((((𝐶 ↾s 𝑆) ∈ V ∧ (Hom
‘ndx) ≠ (Base‘ndx)) ∧ (𝐻 ∈ V ∧ (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V)) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) sSet
〈(Base‘ndx), (𝑇
∩ (Base‘(𝐶
↾s 𝑆)))〉) = (((𝐶 ↾s 𝑆) sSet 〈(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))〉) sSet 〈(Hom
‘ndx), 𝐻〉)) |
75 | 61, 63, 68, 71, 74 | syl22anc 839 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) sSet
〈(Base‘ndx), (𝑇
∩ (Base‘(𝐶
↾s 𝑆)))〉) = (((𝐶 ↾s 𝑆) sSet 〈(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))〉) sSet 〈(Hom
‘ndx), 𝐻〉)) |
76 | | eqid 2737 |
. . . . . . . . . . 11
⊢ ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) ↾s 𝑇) |
77 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘(𝐶
↾s 𝑆)) =
(Base‘(𝐶
↾s 𝑆)) |
78 | 76, 77 | ressval2 16789 |
. . . . . . . . . 10
⊢ ((¬
(Base‘(𝐶
↾s 𝑆))
⊆ 𝑇 ∧ (𝐶 ↾s 𝑆) ∈ V ∧ 𝑇 ∈ V) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet 〈(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))〉)) |
79 | 56, 61, 58, 78 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet 〈(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))〉)) |
80 | 3 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑆 ∈ 𝑊) |
81 | 4 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆) |
82 | | ressabs 16800 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇)) |
83 | 80, 81, 82 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇)) |
84 | 79, 83 | eqtr3d 2779 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet 〈(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))〉) = (𝐶 ↾s 𝑇)) |
85 | 84 | oveq1d 7228 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet 〈(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))〉) sSet 〈(Hom
‘ndx), 𝐻〉) =
((𝐶 ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐻〉)) |
86 | 60, 75, 85 | 3eqtrd 2781 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐻〉)) |
87 | 86 | oveq1d 7228 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉) =
(((𝐶 ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐻〉) sSet
〈(Hom ‘ndx), 𝐽〉)) |
88 | | ovex 7246 |
. . . . . 6
⊢ (𝐶 ↾s 𝑇) ∈ V |
89 | 31 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V) |
90 | | setsabs 16732 |
. . . . . 6
⊢ (((𝐶 ↾s 𝑇) ∈ V ∧ 𝐽 ∈ V) → (((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx),
𝐻〉) sSet 〈(Hom
‘ndx), 𝐽〉) =
((𝐶 ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉)) |
91 | 88, 89, 90 | sylancr 590 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐻〉) sSet 〈(Hom
‘ndx), 𝐽〉) =
((𝐶 ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉)) |
92 | 87, 91 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉) =
((𝐶 ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉)) |
93 | 55, 92 | pm2.61dan 813 |
. . 3
⊢ (𝜑 → ((((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉) =
((𝐶 ↾s
𝑇) sSet 〈(Hom
‘ndx), 𝐽〉)) |
94 | 7, 93 | eqtrd 2777 |
. 2
⊢ (𝜑 → (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾cat
𝐽) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
95 | | eqid 2737 |
. . . 4
⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) |
96 | | rescabs.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ 𝑉) |
97 | 95, 96, 3, 64 | rescval2 17333 |
. . 3
⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
98 | 97 | oveq1d 7228 |
. 2
⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉) ↾cat
𝐽)) |
99 | | eqid 2737 |
. . 3
⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽) |
100 | 99, 96, 5, 6 | rescval2 17333 |
. 2
⊢ (𝜑 → (𝐶 ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
101 | 94, 98, 100 | 3eqtr4d 2787 |
1
⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽)) |