Step | Hyp | Ref
| Expression |
1 | | mptexg 6389 |
. . 3
⊢ (𝐵 ∈ 𝑆 → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V) |
2 | 1 | 3ad2ant2 1076 |
. 2
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V) |
3 | | ffvelrn 6265 |
. . . . . . . . . 10
⊢ (((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)) |
4 | 3 | expcom 450 |
. . . . . . . . 9
⊢ (𝑢 ∈ 𝐵 → ((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))) |
5 | 4 | ralimdv 2946 |
. . . . . . . 8
⊢ (𝑢 ∈ 𝐵 → (∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))) |
6 | 5 | impcom 445 |
. . . . . . 7
⊢
((∀𝑎 ∈
𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)) |
7 | 6 | 3ad2antl3 1218 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)) |
8 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑎 = 𝑠 → (𝐹‘𝑎) = (𝐹‘𝑠)) |
9 | 8 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎)‘𝑢) = ((𝐹‘𝑠)‘𝑢)) |
10 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑎 = 𝑠 → (𝐶‘𝑎) = (𝐶‘𝑠)) |
11 | 9, 10 | eleq12d 2682 |
. . . . . . 7
⊢ (𝑎 = 𝑠 → (((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎) ↔ ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠))) |
12 | 11 | cbvralv 3147 |
. . . . . 6
⊢
(∀𝑎 ∈
𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)) |
13 | 7, 12 | sylib 207 |
. . . . 5
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)) |
14 | | simpl1 1057 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → 𝐴 ∈ 𝑅) |
15 | | mptelixpg 7831 |
. . . . . 6
⊢ (𝐴 ∈ 𝑅 → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ X𝑠 ∈ 𝐴 (𝐶‘𝑠) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠))) |
16 | 14, 15 | syl 17 |
. . . . 5
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ X𝑠 ∈ 𝐴 (𝐶‘𝑠) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠))) |
17 | 13, 16 | mpbird 246 |
. . . 4
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ X𝑠 ∈ 𝐴 (𝐶‘𝑠)) |
18 | | upixp.1 |
. . . . 5
⊢ 𝑋 = X𝑏 ∈ 𝐴 (𝐶‘𝑏) |
19 | | fveq2 6103 |
. . . . . 6
⊢ (𝑏 = 𝑠 → (𝐶‘𝑏) = (𝐶‘𝑠)) |
20 | 19 | cbvixpv 7812 |
. . . . 5
⊢ X𝑏 ∈
𝐴 (𝐶‘𝑏) = X𝑠 ∈ 𝐴 (𝐶‘𝑠) |
21 | 18, 20 | eqtri 2632 |
. . . 4
⊢ 𝑋 = X𝑠 ∈ 𝐴 (𝐶‘𝑠) |
22 | 17, 21 | syl6eleqr 2699 |
. . 3
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋) |
23 | | eqid 2610 |
. . 3
⊢ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) |
24 | 22, 23 | fmptd 6292 |
. 2
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋) |
25 | | nfv 1830 |
. . . 4
⊢
Ⅎ𝑎 𝐴 ∈ 𝑅 |
26 | | nfv 1830 |
. . . 4
⊢
Ⅎ𝑎 𝐵 ∈ 𝑆 |
27 | | nfra1 2925 |
. . . 4
⊢
Ⅎ𝑎∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) |
28 | 25, 26, 27 | nf3an 1819 |
. . 3
⊢
Ⅎ𝑎(𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) |
29 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑠 = 𝑎 → (𝐹‘𝑠) = (𝐹‘𝑎)) |
30 | 29 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝑠 = 𝑎 → ((𝐹‘𝑠)‘𝑢) = ((𝐹‘𝑎)‘𝑢)) |
31 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) |
32 | | fvex 6113 |
. . . . . . . 8
⊢ ((𝐹‘𝑠)‘𝑢) ∈ V |
33 | 30, 31, 32 | fvmpt3i 6196 |
. . . . . . 7
⊢ (𝑎 ∈ 𝐴 → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎) = ((𝐹‘𝑎)‘𝑢)) |
34 | 33 | adantl 481 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎) = ((𝐹‘𝑎)‘𝑢)) |
35 | 34 | mpteq2dv 4673 |
. . . . 5
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑢 ∈ 𝐵 ↦ ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎)) = (𝑢 ∈ 𝐵 ↦ ((𝐹‘𝑎)‘𝑢))) |
36 | 22 | adantlr 747 |
. . . . . 6
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋) |
37 | | eqidd 2611 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))) |
38 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑤 = 𝑎 → (𝑥‘𝑤) = (𝑥‘𝑎)) |
39 | 38 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (𝑤 = 𝑎 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎))) |
40 | | upixp.2 |
. . . . . . . 8
⊢ 𝑃 = (𝑤 ∈ 𝐴 ↦ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤))) |
41 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝐶‘𝑏) ∈ V |
42 | 41 | rgenw 2908 |
. . . . . . . . . . 11
⊢
∀𝑏 ∈
𝐴 (𝐶‘𝑏) ∈ V |
43 | | ixpexg 7818 |
. . . . . . . . . . 11
⊢
(∀𝑏 ∈
𝐴 (𝐶‘𝑏) ∈ V → X𝑏 ∈
𝐴 (𝐶‘𝑏) ∈ V) |
44 | 42, 43 | ax-mp 5 |
. . . . . . . . . 10
⊢ X𝑏 ∈
𝐴 (𝐶‘𝑏) ∈ V |
45 | 18, 44 | eqeltri 2684 |
. . . . . . . . 9
⊢ 𝑋 ∈ V |
46 | 45 | mptex 6390 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) ∈ V |
47 | 39, 40, 46 | fvmpt3i 6196 |
. . . . . . 7
⊢ (𝑎 ∈ 𝐴 → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎))) |
48 | 47 | adantl 481 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎))) |
49 | | fveq1 6102 |
. . . . . 6
⊢ (𝑥 = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) → (𝑥‘𝑎) = ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎)) |
50 | 36, 37, 48, 49 | fmptco 6303 |
. . . . 5
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))) = (𝑢 ∈ 𝐵 ↦ ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎))) |
51 | | rsp 2913 |
. . . . . . . 8
⊢
(∀𝑎 ∈
𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎))) |
52 | 51 | 3ad2ant3 1077 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎))) |
53 | 52 | imp 444 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) |
54 | 53 | feqmptd 6159 |
. . . . 5
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = (𝑢 ∈ 𝐵 ↦ ((𝐹‘𝑎)‘𝑢))) |
55 | 35, 50, 54 | 3eqtr4rd 2655 |
. . . 4
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) |
56 | 55 | ex 449 |
. . 3
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))) |
57 | 28, 56 | ralrimi 2940 |
. 2
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) |
58 | | simprl 790 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → ℎ:𝐵⟶𝑋) |
59 | 58 | feqmptd 6159 |
. . . . 5
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → ℎ = (𝑢 ∈ 𝐵 ↦ (ℎ‘𝑢))) |
60 | | simplrr 797 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) |
61 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑠 → (𝑃‘𝑎) = (𝑃‘𝑠)) |
62 | 61 | coeq1d 5205 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑠 → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑠) ∘ ℎ)) |
63 | 8, 62 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ))) |
64 | 63 | rspccva 3281 |
. . . . . . . . . . 11
⊢
((∀𝑎 ∈
𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ∧ 𝑠 ∈ 𝐴) → (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ)) |
65 | 60, 64 | sylan 487 |
. . . . . . . . . 10
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ)) |
66 | 65 | fveq1d 6105 |
. . . . . . . . 9
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘𝑠)‘𝑢) = (((𝑃‘𝑠) ∘ ℎ)‘𝑢)) |
67 | | fvco3 6185 |
. . . . . . . . . . 11
⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢))) |
68 | 58, 67 | sylan 487 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢))) |
69 | 68 | adantr 480 |
. . . . . . . . 9
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢))) |
70 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑠 → (𝑥‘𝑤) = (𝑥‘𝑠)) |
71 | 70 | mpteq2dv 4673 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑠 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))) |
72 | 71, 40, 46 | fvmpt3i 6196 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ 𝐴 → (𝑃‘𝑠) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))) |
73 | 72 | adantl 481 |
. . . . . . . . . . 11
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → (𝑃‘𝑠) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))) |
74 | 73 | fveq1d 6105 |
. . . . . . . . . 10
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝑃‘𝑠)‘(ℎ‘𝑢)) = ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢))) |
75 | | ffvelrn 6265 |
. . . . . . . . . . . . 13
⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ 𝑋) |
76 | 58, 75 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ 𝑋) |
77 | | fveq1 6102 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (ℎ‘𝑢) → (𝑥‘𝑠) = ((ℎ‘𝑢)‘𝑠)) |
78 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)) |
79 | | fvex 6113 |
. . . . . . . . . . . . 13
⊢ (𝑥‘𝑠) ∈ V |
80 | 77, 78, 79 | fvmpt3i 6196 |
. . . . . . . . . . . 12
⊢ ((ℎ‘𝑢) ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠)) |
81 | 76, 80 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠)) |
82 | 81 | adantr 480 |
. . . . . . . . . 10
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠)) |
83 | 74, 82 | eqtrd 2644 |
. . . . . . . . 9
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝑃‘𝑠)‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠)) |
84 | 66, 69, 83 | 3eqtrd 2648 |
. . . . . . . 8
⊢
(((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘𝑠)‘𝑢) = ((ℎ‘𝑢)‘𝑠)) |
85 | 84 | mpteq2dva 4672 |
. . . . . . 7
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (𝑠 ∈ 𝐴 ↦ ((ℎ‘𝑢)‘𝑠))) |
86 | 76, 18 | syl6eleq 2698 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ X𝑏 ∈ 𝐴 (𝐶‘𝑏)) |
87 | | ixpfn 7800 |
. . . . . . . . 9
⊢ ((ℎ‘𝑢) ∈ X𝑏 ∈ 𝐴 (𝐶‘𝑏) → (ℎ‘𝑢) Fn 𝐴) |
88 | 86, 87 | syl 17 |
. . . . . . . 8
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) Fn 𝐴) |
89 | | dffn5 6151 |
. . . . . . . 8
⊢ ((ℎ‘𝑢) Fn 𝐴 ↔ (ℎ‘𝑢) = (𝑠 ∈ 𝐴 ↦ ((ℎ‘𝑢)‘𝑠))) |
90 | 88, 89 | sylib 207 |
. . . . . . 7
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) = (𝑠 ∈ 𝐴 ↦ ((ℎ‘𝑢)‘𝑠))) |
91 | 85, 90 | eqtr4d 2647 |
. . . . . 6
⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (ℎ‘𝑢)) |
92 | 91 | mpteq2dva 4672 |
. . . . 5
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) = (𝑢 ∈ 𝐵 ↦ (ℎ‘𝑢))) |
93 | 59, 92 | eqtr4d 2647 |
. . . 4
⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))) |
94 | 93 | ex 449 |
. . 3
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) |
95 | 94 | alrimiv 1842 |
. 2
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∀ℎ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) |
96 | | feq1 5939 |
. . . 4
⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → (ℎ:𝐵⟶𝑋 ↔ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋)) |
97 | | coeq2 5202 |
. . . . . 6
⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) |
98 | 97 | eqeq2d 2620 |
. . . . 5
⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))) |
99 | 98 | ralbidv 2969 |
. . . 4
⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → (∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))) |
100 | 96, 99 | anbi12d 743 |
. . 3
⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) ↔ ((𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))))) |
101 | 100 | eqeu 3344 |
. 2
⊢ (((𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V ∧ ((𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) ∧ ∀ℎ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) |
102 | 2, 24, 57, 95, 101 | syl121anc 1323 |
1
⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) |