Step | Hyp | Ref
| Expression |
1 | | fsplitfpar.s |
. . . . . . . . . 10
⊢ 𝑆 = (◡(1st ↾ I ) ↾ 𝐴) |
2 | | fsplit 7841 |
. . . . . . . . . . 11
⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) |
3 | 2 | reseq1i 5822 |
. . . . . . . . . 10
⊢ (◡(1st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) ↾ 𝐴) |
4 | 1, 3 | eqtri 2762 |
. . . . . . . . 9
⊢ 𝑆 = ((𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) ↾ 𝐴) |
5 | 4 | fveq1i 6678 |
. . . . . . . 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 6696 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ 〈𝑥, 𝑥〉)‘𝑎)) |
8 | | eqidd 2740 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐴 → (𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) = (𝑥 ∈ V ↦ 〈𝑥, 𝑥〉)) |
9 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎) |
10 | 9, 9 | opeq12d 4770 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑎 → 〈𝑥, 𝑥〉 = 〈𝑎, 𝑎〉) |
11 | 10 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = 𝑎) → 〈𝑥, 𝑥〉 = 〈𝑎, 𝑎〉) |
12 | | elex 3417 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ V) |
13 | | opex 5323 |
. . . . . . . . . . 11
⊢
〈𝑎, 𝑎〉 ∈ V |
14 | 13 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐴 → 〈𝑎, 𝑎〉 ∈ V) |
15 | 8, 11, 12, 14 | fvmptd 6785 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ V ↦ 〈𝑥, 𝑥〉)‘𝑎) = 〈𝑎, 𝑎〉) |
16 | 7, 15 | eqtrd 2774 |
. . . . . . . 8
⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) ↾ 𝐴)‘𝑎) = 〈𝑎, 𝑎〉) |
17 | 16 | adantl 485 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) ↾ 𝐴)‘𝑎) = 〈𝑎, 𝑎〉) |
18 | 6, 17 | eqtrd 2774 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = 〈𝑎, 𝑎〉) |
19 | 18 | fveq2d 6681 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = (𝐻‘〈𝑎, 𝑎〉)) |
20 | | df-ov 7176 |
. . . . . 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 7840 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉)) |
23 | 22 | adantr 484 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉)) |
24 | | fveq2 6677 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) |
25 | 24 | adantr 484 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹‘𝑥) = (𝐹‘𝑎)) |
26 | | fveq2 6677 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑎 → (𝐺‘𝑦) = (𝐺‘𝑎)) |
27 | 26 | adantl 485 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐺‘𝑦) = (𝐺‘𝑎)) |
28 | 25, 27 | opeq12d 4770 |
. . . . . . . 8
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → 〈(𝐹‘𝑥), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
29 | 28 | adantl 485 |
. . . . . . 7
⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑎)) → 〈(𝐹‘𝑥), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
30 | | simpr 488 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴) |
31 | | opex 5323 |
. . . . . . . 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 7320 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑎𝐻𝑎) = 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
34 | 20, 33 | eqtr3id 2788 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘〈𝑎, 𝑎〉) = 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
35 | 19, 34 | eqtrd 2774 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
36 | | eqid 2739 |
. . . . . . . . . 10
⊢ (𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) = (𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) |
37 | 36 | fnmpt 6478 |
. . . . . . . . 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 3902 |
. . . . . . . 8
⊢ 𝐴 ⊆ V |
41 | | fnssres 6460 |
. . . . . . . 8
⊢ (((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) Fn 𝐴) |
42 | 39, 40, 41 | mp2an 692 |
. . . . . . 7
⊢ ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) Fn 𝐴 |
43 | | fsplit 7841 |
. . . . . . . . . 10
⊢ ◡(1st ↾ I ) = (𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) |
44 | 43 | reseq1i 5822 |
. . . . . . . . 9
⊢ (◡(1st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) |
45 | 1, 44 | eqtri 2762 |
. . . . . . . 8
⊢ 𝑆 = ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) |
46 | 45 | fneq1i 6436 |
. . . . . . 7
⊢ (𝑆 Fn 𝐴 ↔ ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) Fn 𝐴) |
47 | 42, 46 | mpbir 234 |
. . . . . 6
⊢ 𝑆 Fn 𝐴 |
48 | 47 | a1i 11 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴) |
49 | | fvco2 6768 |
. . . . 5
⊢ ((𝑆 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎))) |
50 | 48, 49 | sylan 583 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎))) |
51 | | fveq2 6677 |
. . . . . . 7
⊢ (𝑥 = 𝑎 → (𝐺‘𝑥) = (𝐺‘𝑎)) |
52 | 24, 51 | opeq12d 4770 |
. . . . . 6
⊢ (𝑥 = 𝑎 → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
53 | | eqid 2739 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = (𝑥 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉) |
54 | 52, 53, 31 | fvmpt 6778 |
. . . . 5
⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉)‘𝑎) = 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
55 | 54 | adantl 485 |
. . . 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 2784 |
. . 3
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉)‘𝑎)) |
57 | 56 | ralrimiva 3097 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉)‘𝑎)) |
58 | | opex 5323 |
. . . . . . . 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 3104 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 〈(𝐹‘𝑥), (𝐺‘𝑦)〉 ∈ V) |
61 | | eqid 2739 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) |
62 | 61 | fnmpo 7795 |
. . . . . 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 6432 |
. . . . 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 260 |
. . . 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 3097 |
. . . . . . 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 589 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) Fn 𝐴) |
70 | 69, 46 | sylibr 237 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴) |
71 | 45 | rneqi 5781 |
. . . . . 6
⊢ ran 𝑆 = ran ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) |
72 | | mptima 5916 |
. . . . . . 7
⊢ ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ 〈𝑎, 𝑎〉) |
73 | | df-ima 5539 |
. . . . . . 7
⊢ ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) “ 𝐴) = ran ((𝑎 ∈ V ↦ 〈𝑎, 𝑎〉) ↾ 𝐴) |
74 | | eqid 2739 |
. . . . . . . 8
⊢ (𝑎 ∈ (V ∩ 𝐴) ↦ 〈𝑎, 𝑎〉) = (𝑎 ∈ (V ∩ 𝐴) ↦ 〈𝑎, 𝑎〉) |
75 | 74 | rnmpt 5799 |
. . . . . . 7
⊢ ran
(𝑎 ∈ (V ∩ 𝐴) ↦ 〈𝑎, 𝑎〉) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = 〈𝑎, 𝑎〉} |
76 | 72, 73, 75 | 3eqtr3i 2770 |
. . . . . 6
⊢ ran
((𝑎 ∈ V ↦
〈𝑎, 𝑎〉) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = 〈𝑎, 𝑎〉} |
77 | 71, 76 | eqtri 2762 |
. . . . 5
⊢ ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = 〈𝑎, 𝑎〉} |
78 | | elinel2 4087 |
. . . . . . . . 9
⊢ (𝑎 ∈ (V ∩ 𝐴) → 𝑎 ∈ 𝐴) |
79 | | simpl 486 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = 〈𝑎, 𝑎〉) → 𝑎 ∈ 𝐴) |
80 | 79, 79 | opelxpd 5564 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = 〈𝑎, 𝑎〉) → 〈𝑎, 𝑎〉 ∈ (𝐴 × 𝐴)) |
81 | | eleq1 2821 |
. . . . . . . . . . . 12
⊢ (𝑝 = 〈𝑎, 𝑎〉 → (𝑝 ∈ (𝐴 × 𝐴) ↔ 〈𝑎, 𝑎〉 ∈ (𝐴 × 𝐴))) |
82 | 81 | adantl 485 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = 〈𝑎, 𝑎〉) → (𝑝 ∈ (𝐴 × 𝐴) ↔ 〈𝑎, 𝑎〉 ∈ (𝐴 × 𝐴))) |
83 | 80, 82 | mpbird 260 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = 〈𝑎, 𝑎〉) → 𝑝 ∈ (𝐴 × 𝐴)) |
84 | 83 | ex 416 |
. . . . . . . . 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 3191 |
. . . . . . 7
⊢
(∃𝑎 ∈ (V
∩ 𝐴)𝑝 = 〈𝑎, 𝑎〉 → 𝑝 ∈ (𝐴 × 𝐴)) |
87 | 86 | abssi 3960 |
. . . . . 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 3926 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴)) |
90 | | fnco 6454 |
. . . 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 5323 |
. . . . . 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 3097 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ V) |
95 | 53 | fnmpt 6478 |
. . . 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 6812 |
. . 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 587 |
. 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 260 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |