Step | Hyp | Ref
| Expression |
1 | | fsplitfpar.s |
. . . . . . . . . 10
⊢ 𝑆 = (◡(1st ↾ I ) ↾ 𝐴) |
2 | | fsplit 8103 |
. . . . . . . . . . 11
⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) |
3 | 2 | reseq1i 5978 |
. . . . . . . . . 10
⊢ (◡(1st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴) |
4 | 1, 3 | eqtri 2761 |
. . . . . . . . 9
⊢ 𝑆 = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴) |
5 | 4 | fveq1i 6893 |
. . . . . . . 8
⊢ (𝑆‘𝑎) = (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) |
6 | 5 | a1i 11 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎)) |
7 | | fvres 6911 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎)) |
8 | | eqidd 2734 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐴 → (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)) |
9 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎) |
10 | 9, 9 | opeq12d 4882 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑎 → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩) |
11 | 10 | adantl 483 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = 𝑎) → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩) |
12 | | elex 3493 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ V) |
13 | | opex 5465 |
. . . . . . . . . . 11
⊢
⟨𝑎, 𝑎⟩ ∈ V |
14 | 13 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐴 → ⟨𝑎, 𝑎⟩ ∈ V) |
15 | 8, 11, 12, 14 | fvmptd 7006 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎) = ⟨𝑎, 𝑎⟩) |
16 | 7, 15 | eqtrd 2773 |
. . . . . . . 8
⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩) |
17 | 16 | adantl 483 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩) |
18 | 6, 17 | eqtrd 2773 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = ⟨𝑎, 𝑎⟩) |
19 | 18 | fveq2d 6896 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = (𝐻‘⟨𝑎, 𝑎⟩)) |
20 | | df-ov 7412 |
. . . . . 6
⊢ (𝑎𝐻𝑎) = (𝐻‘⟨𝑎, 𝑎⟩) |
21 | | fsplitfpar.h |
. . . . . . . . 9
⊢ 𝐻 = ((◡(1st ↾ (V × V))
∘ (𝐹 ∘
(1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V))
∘ (𝐺 ∘
(2nd ↾ (V × V))))) |
22 | 21 | fpar 8102 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)) |
23 | 22 | adantr 482 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)) |
24 | | fveq2 6892 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) |
25 | 24 | adantr 482 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹‘𝑥) = (𝐹‘𝑎)) |
26 | | fveq2 6892 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑎 → (𝐺‘𝑦) = (𝐺‘𝑎)) |
27 | 26 | adantl 483 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐺‘𝑦) = (𝐺‘𝑎)) |
28 | 25, 27 | opeq12d 4882 |
. . . . . . . 8
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
29 | 28 | adantl 483 |
. . . . . . 7
⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑎)) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
30 | | simpr 486 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴) |
31 | | opex 5465 |
. . . . . . . 8
⊢
⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V |
32 | 31 | a1i 11 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V) |
33 | 23, 29, 30, 30, 32 | ovmpod 7560 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑎𝐻𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
34 | 20, 33 | eqtr3id 2787 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘⟨𝑎, 𝑎⟩) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
35 | 19, 34 | eqtrd 2773 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
36 | | eqid 2733 |
. . . . . . . . . 10
⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) |
37 | 36 | fnmpt 6691 |
. . . . . . . . 9
⊢
(∀𝑎 ∈ V
⟨𝑎, 𝑎⟩ ∈ V → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V) |
38 | 13 | a1i 11 |
. . . . . . . . 9
⊢ (𝑎 ∈ V → ⟨𝑎, 𝑎⟩ ∈ V) |
39 | 37, 38 | mprg 3068 |
. . . . . . . 8
⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V |
40 | | ssv 4007 |
. . . . . . . 8
⊢ 𝐴 ⊆ V |
41 | | fnssres 6674 |
. . . . . . . 8
⊢ (((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴) |
42 | 39, 40, 41 | mp2an 691 |
. . . . . . 7
⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴 |
43 | | fsplit 8103 |
. . . . . . . . . 10
⊢ ◡(1st ↾ I ) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) |
44 | 43 | reseq1i 5978 |
. . . . . . . . 9
⊢ (◡(1st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) |
45 | 1, 44 | eqtri 2761 |
. . . . . . . 8
⊢ 𝑆 = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) |
46 | 45 | fneq1i 6647 |
. . . . . . 7
⊢ (𝑆 Fn 𝐴 ↔ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴) |
47 | 42, 46 | mpbir 230 |
. . . . . 6
⊢ 𝑆 Fn 𝐴 |
48 | 47 | a1i 11 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴) |
49 | | fvco2 6989 |
. . . . 5
⊢ ((𝑆 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎))) |
50 | 48, 49 | sylan 581 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎))) |
51 | | fveq2 6892 |
. . . . . . 7
⊢ (𝑥 = 𝑎 → (𝐺‘𝑥) = (𝐺‘𝑎)) |
52 | 24, 51 | opeq12d 4882 |
. . . . . 6
⊢ (𝑥 = 𝑎 → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
53 | | eqid 2733 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) |
54 | 52, 53, 31 | fvmpt 6999 |
. . . . 5
⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
55 | 54 | adantl 483 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
56 | 35, 50, 55 | 3eqtr4d 2783 |
. . 3
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎)) |
57 | 56 | ralrimiva 3147 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎)) |
58 | | opex 5465 |
. . . . . . . 8
⊢
⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V |
59 | 58 | a1i 11 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V) |
60 | 59 | ralrimivva 3201 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V) |
61 | | eqid 2733 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) |
62 | 61 | fnmpo 8055 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) Fn (𝐴 × 𝐴)) |
63 | 60, 62 | syl 17 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) Fn (𝐴 × 𝐴)) |
64 | 22 | fneq1d 6643 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 Fn (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) Fn (𝐴 × 𝐴))) |
65 | 63, 64 | mpbird 257 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐻 Fn (𝐴 × 𝐴)) |
66 | 13 | a1i 11 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ V) → ⟨𝑎, 𝑎⟩ ∈ V) |
67 | 66 | ralrimiva 3147 |
. . . . . . 7
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V) |
68 | 67, 37 | syl 17 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V) |
69 | 68, 40, 41 | sylancl 587 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴) |
70 | 69, 46 | sylibr 233 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴) |
71 | 45 | rneqi 5937 |
. . . . . 6
⊢ ran 𝑆 = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) |
72 | | mptima 6072 |
. . . . . . 7
⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) |
73 | | df-ima 5690 |
. . . . . . 7
⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) |
74 | | eqid 2733 |
. . . . . . . 8
⊢ (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) |
75 | 74 | rnmpt 5955 |
. . . . . . 7
⊢ ran
(𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} |
76 | 72, 73, 75 | 3eqtr3i 2769 |
. . . . . 6
⊢ ran
((𝑎 ∈ V ↦
⟨𝑎, 𝑎⟩) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} |
77 | 71, 76 | eqtri 2761 |
. . . . 5
⊢ ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} |
78 | | elinel2 4197 |
. . . . . . . . 9
⊢ (𝑎 ∈ (V ∩ 𝐴) → 𝑎 ∈ 𝐴) |
79 | | simpl 484 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → 𝑎 ∈ 𝐴) |
80 | 79, 79 | opelxpd 5716 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)) |
81 | | eleq1 2822 |
. . . . . . . . . . . 12
⊢ (𝑝 = ⟨𝑎, 𝑎⟩ → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))) |
82 | 81 | adantl 483 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))) |
83 | 80, 82 | mpbird 257 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → 𝑝 ∈ (𝐴 × 𝐴)) |
84 | 83 | ex 414 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐴 → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))) |
85 | 78, 84 | syl 17 |
. . . . . . . 8
⊢ (𝑎 ∈ (V ∩ 𝐴) → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))) |
86 | 85 | rexlimiv 3149 |
. . . . . . 7
⊢
(∃𝑎 ∈ (V
∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)) |
87 | 86 | abssi 4068 |
. . . . . 6
⊢ {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} ⊆ (𝐴 × 𝐴) |
88 | 87 | a1i 11 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} ⊆ (𝐴 × 𝐴)) |
89 | 77, 88 | eqsstrid 4031 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴)) |
90 | | fnco 6668 |
. . . 4
⊢ ((𝐻 Fn (𝐴 × 𝐴) ∧ 𝑆 Fn 𝐴 ∧ ran 𝑆 ⊆ (𝐴 × 𝐴)) → (𝐻 ∘ 𝑆) Fn 𝐴) |
91 | 65, 70, 89, 90 | syl3anc 1372 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) Fn 𝐴) |
92 | | opex 5465 |
. . . . . 6
⊢
⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V |
93 | 92 | a1i 11 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V) |
94 | 93 | ralrimiva 3147 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V) |
95 | 53 | fnmpt 6691 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴) |
96 | 94, 95 | syl 17 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴) |
97 | | eqfnfv 7033 |
. . 3
⊢ (((𝐻 ∘ 𝑆) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴) → ((𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ↔ ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))) |
98 | 91, 96, 97 | syl2anc 585 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ↔ ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))) |
99 | 57, 98 | mpbird 257 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) |