Step | Hyp | Ref
| Expression |
1 | | fnfun 6603 |
. . . . 5
⊢ (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹) |
2 | | 1stconst 8033 |
. . . . . 6
⊢ (𝐶 ∈ 𝐵 → (1st ↾ (V ×
{𝐶})):(V × {𝐶})–1-1-onto→V) |
3 | | dff1o3 6791 |
. . . . . . 7
⊢
((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V
↔ ((1st ↾ (V × {𝐶})):(V × {𝐶})–onto→V ∧ Fun ◡(1st ↾ (V × {𝐶})))) |
4 | 3 | simprbi 498 |
. . . . . 6
⊢
((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V
→ Fun ◡(1st ↾ (V
× {𝐶}))) |
5 | 2, 4 | syl 17 |
. . . . 5
⊢ (𝐶 ∈ 𝐵 → Fun ◡(1st ↾ (V × {𝐶}))) |
6 | | funco 6542 |
. . . . 5
⊢ ((Fun
𝐹 ∧ Fun ◡(1st ↾ (V × {𝐶}))) → Fun (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))) |
7 | 1, 5, 6 | syl2an 597 |
. . . 4
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → Fun (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))) |
8 | | dmco 6207 |
. . . . 5
⊢ dom
(𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) = (◡◡(1st ↾ (V × {𝐶})) “ dom 𝐹) |
9 | | fndm 6606 |
. . . . . . . 8
⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵)) |
10 | 9 | adantr 482 |
. . . . . . 7
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → dom 𝐹 = (𝐴 × 𝐵)) |
11 | 10 | imaeq2d 6014 |
. . . . . 6
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (◡◡(1st ↾ (V × {𝐶})) “ dom 𝐹) = (◡◡(1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵))) |
12 | | imacnvcnv 6159 |
. . . . . . . . 9
⊢ (◡◡(1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ((1st ↾ (V ×
{𝐶})) “ (𝐴 × 𝐵)) |
13 | | df-ima 5647 |
. . . . . . . . 9
⊢
((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran ((1st ↾ (V ×
{𝐶})) ↾ (𝐴 × 𝐵)) |
14 | | resres 5951 |
. . . . . . . . . 10
⊢
((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V ×
{𝐶}) ∩ (𝐴 × 𝐵))) |
15 | 14 | rneqi 5893 |
. . . . . . . . 9
⊢ ran
((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = ran (1st ↾ ((V ×
{𝐶}) ∩ (𝐴 × 𝐵))) |
16 | 12, 13, 15 | 3eqtri 2769 |
. . . . . . . 8
⊢ (◡◡(1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran (1st ↾ ((V ×
{𝐶}) ∩ (𝐴 × 𝐵))) |
17 | | inxp 5789 |
. . . . . . . . . . . . 13
⊢ ((V
× {𝐶}) ∩ (𝐴 × 𝐵)) = ((V ∩ 𝐴) × ({𝐶} ∩ 𝐵)) |
18 | | incom 4162 |
. . . . . . . . . . . . . . 15
⊢ (V ∩
𝐴) = (𝐴 ∩ V) |
19 | | inv1 4355 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∩ V) = 𝐴 |
20 | 18, 19 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢ (V ∩
𝐴) = 𝐴 |
21 | 20 | xpeq1i 5660 |
. . . . . . . . . . . . 13
⊢ ((V ∩
𝐴) × ({𝐶} ∩ 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵)) |
22 | 17, 21 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ ((V
× {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵)) |
23 | | snssi 4769 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ∈ 𝐵 → {𝐶} ⊆ 𝐵) |
24 | | df-ss 3928 |
. . . . . . . . . . . . . 14
⊢ ({𝐶} ⊆ 𝐵 ↔ ({𝐶} ∩ 𝐵) = {𝐶}) |
25 | 23, 24 | sylib 217 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ 𝐵 → ({𝐶} ∩ 𝐵) = {𝐶}) |
26 | 25 | xpeq2d 5664 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ 𝐵 → (𝐴 × ({𝐶} ∩ 𝐵)) = (𝐴 × {𝐶})) |
27 | 22, 26 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐵 → ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × {𝐶})) |
28 | 27 | reseq2d 5938 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐵 → (1st ↾ ((V ×
{𝐶}) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × {𝐶}))) |
29 | 28 | rneqd 5894 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐵 → ran (1st ↾ ((V
× {𝐶}) ∩ (𝐴 × 𝐵))) = ran (1st ↾ (𝐴 × {𝐶}))) |
30 | | 1stconst 8033 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐵 → (1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto→𝐴) |
31 | | f1ofo 6792 |
. . . . . . . . . 10
⊢
((1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto→𝐴 → (1st ↾
(𝐴 × {𝐶})):(𝐴 × {𝐶})–onto→𝐴) |
32 | | forn 6760 |
. . . . . . . . . 10
⊢
((1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–onto→𝐴 → ran (1st ↾ (𝐴 × {𝐶})) = 𝐴) |
33 | 30, 31, 32 | 3syl 18 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐵 → ran (1st ↾ (𝐴 × {𝐶})) = 𝐴) |
34 | 29, 33 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝐶 ∈ 𝐵 → ran (1st ↾ ((V
× {𝐶}) ∩ (𝐴 × 𝐵))) = 𝐴) |
35 | 16, 34 | eqtrid 2789 |
. . . . . . 7
⊢ (𝐶 ∈ 𝐵 → (◡◡(1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴) |
36 | 35 | adantl 483 |
. . . . . 6
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (◡◡(1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴) |
37 | 11, 36 | eqtrd 2777 |
. . . . 5
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (◡◡(1st ↾ (V × {𝐶})) “ dom 𝐹) = 𝐴) |
38 | 8, 37 | eqtrid 2789 |
. . . 4
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → dom (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) = 𝐴) |
39 | | curry2.1 |
. . . . . 6
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) |
40 | 39 | fneq1i 6600 |
. . . . 5
⊢ (𝐺 Fn 𝐴 ↔ (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) Fn 𝐴) |
41 | | df-fn 6500 |
. . . . 5
⊢ ((𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) Fn 𝐴 ↔ (Fun (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ∧ dom (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) = 𝐴)) |
42 | 40, 41 | bitri 275 |
. . . 4
⊢ (𝐺 Fn 𝐴 ↔ (Fun (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ∧ dom (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) = 𝐴)) |
43 | 7, 38, 42 | sylanbrc 584 |
. . 3
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 Fn 𝐴) |
44 | | dffn5 6902 |
. . 3
⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) |
45 | 43, 44 | sylib 217 |
. 2
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) |
46 | 39 | fveq1i 6844 |
. . . . 5
⊢ (𝐺‘𝑥) = ((𝐹 ∘ ◡(1st ↾ (V × {𝐶})))‘𝑥) |
47 | | dff1o4 6793 |
. . . . . . . . 9
⊢
((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V
↔ ((1st ↾ (V × {𝐶})) Fn (V × {𝐶}) ∧ ◡(1st ↾ (V × {𝐶})) Fn V)) |
48 | 2, 47 | sylib 217 |
. . . . . . . 8
⊢ (𝐶 ∈ 𝐵 → ((1st ↾ (V ×
{𝐶})) Fn (V × {𝐶}) ∧ ◡(1st ↾ (V × {𝐶})) Fn V)) |
49 | 48 | simprd 497 |
. . . . . . 7
⊢ (𝐶 ∈ 𝐵 → ◡(1st ↾ (V × {𝐶})) Fn V) |
50 | | vex 3450 |
. . . . . . 7
⊢ 𝑥 ∈ V |
51 | | fvco2 6939 |
. . . . . . 7
⊢ ((◡(1st ↾ (V × {𝐶})) Fn V ∧ 𝑥 ∈ V) → ((𝐹 ∘ ◡(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘(◡(1st ↾ (V × {𝐶}))‘𝑥))) |
52 | 49, 50, 51 | sylancl 587 |
. . . . . 6
⊢ (𝐶 ∈ 𝐵 → ((𝐹 ∘ ◡(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘(◡(1st ↾ (V × {𝐶}))‘𝑥))) |
53 | 52 | ad2antlr 726 |
. . . . 5
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ ◡(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘(◡(1st ↾ (V × {𝐶}))‘𝑥))) |
54 | 46, 53 | eqtrid 2789 |
. . . 4
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘(◡(1st ↾ (V × {𝐶}))‘𝑥))) |
55 | 2 | adantr 482 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (1st ↾ (V ×
{𝐶})):(V × {𝐶})–1-1-onto→V) |
56 | 50 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ V) |
57 | | snidg 4621 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ {𝐶}) |
58 | 57 | adantr 482 |
. . . . . . . . . 10
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ {𝐶}) |
59 | 56, 58 | opelxpd 5672 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶})) |
60 | 55, 59 | jca 513 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((1st ↾ (V ×
{𝐶})):(V × {𝐶})–1-1-onto→V
∧ ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}))) |
61 | 50 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ 𝐵 → 𝑥 ∈ V) |
62 | 61, 57 | opelxpd 5672 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐵 → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶})) |
63 | 62 | fvresd 6863 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐵 → ((1st ↾ (V ×
{𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩)) |
64 | 63 | adantr 482 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((1st ↾ (V ×
{𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩)) |
65 | | op1stg 7934 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥) |
66 | 65 | ancoms 460 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥) |
67 | 64, 66 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((1st ↾ (V ×
{𝐶}))‘⟨𝑥, 𝐶⟩) = 𝑥) |
68 | | f1ocnvfv 7225 |
. . . . . . . 8
⊢
(((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V
∧ ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶})) → (((1st
↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = 𝑥 → (◡(1st ↾ (V × {𝐶}))‘𝑥) = ⟨𝑥, 𝐶⟩)) |
69 | 60, 67, 68 | sylc 65 |
. . . . . . 7
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (◡(1st ↾ (V × {𝐶}))‘𝑥) = ⟨𝑥, 𝐶⟩) |
70 | 69 | fveq2d 6847 |
. . . . . 6
⊢ ((𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘(◡(1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩)) |
71 | 70 | adantll 713 |
. . . . 5
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(◡(1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩)) |
72 | | df-ov 7361 |
. . . . 5
⊢ (𝑥𝐹𝐶) = (𝐹‘⟨𝑥, 𝐶⟩) |
73 | 71, 72 | eqtr4di 2795 |
. . . 4
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(◡(1st ↾ (V × {𝐶}))‘𝑥)) = (𝑥𝐹𝐶)) |
74 | 54, 73 | eqtrd 2777 |
. . 3
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝑥𝐹𝐶)) |
75 | 74 | mpteq2dva 5206 |
. 2
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) |
76 | 45, 75 | eqtrd 2777 |
1
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) |