Proof of Theorem curry1
Step | Hyp | Ref
| Expression |
1 | | fnfun 5988 |
. . . . 5
⊢ (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹) |
2 | | 2ndconst 7266 |
. . . . . 6
⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V) |
3 | | dff1o3 6143 |
. . . . . . 7
⊢
((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun ◡(2nd ↾ ({𝐶} × V)))) |
4 | 3 | simprbi 480 |
. . . . . 6
⊢
((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
→ Fun ◡(2nd ↾
({𝐶} ×
V))) |
5 | 2, 4 | syl 17 |
. . . . 5
⊢ (𝐶 ∈ 𝐴 → Fun ◡(2nd ↾ ({𝐶} × V))) |
6 | | funco 5928 |
. . . . 5
⊢ ((Fun
𝐹 ∧ Fun ◡(2nd ↾ ({𝐶} × V))) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))) |
7 | 1, 5, 6 | syl2an 494 |
. . . 4
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))) |
8 | | dmco 5643 |
. . . . 5
⊢ dom
(𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) |
9 | | fndm 5990 |
. . . . . . . 8
⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵)) |
10 | 9 | adantr 481 |
. . . . . . 7
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵)) |
11 | 10 | imaeq2d 5466 |
. . . . . 6
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))) |
12 | | imacnvcnv 5599 |
. . . . . . . . 9
⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) |
13 | | df-ima 5127 |
. . . . . . . . 9
⊢
((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) |
14 | | resres 5409 |
. . . . . . . . . 10
⊢
((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) |
15 | 14 | rneqi 5352 |
. . . . . . . . 9
⊢ ran
((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) |
16 | 12, 13, 15 | 3eqtri 2648 |
. . . . . . . 8
⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) |
17 | | inxp 5254 |
. . . . . . . . . . . . 13
⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) |
18 | | incom 3805 |
. . . . . . . . . . . . . . 15
⊢ (V ∩
𝐵) = (𝐵 ∩ V) |
19 | | inv1 3970 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∩ V) = 𝐵 |
20 | 18, 19 | eqtri 2644 |
. . . . . . . . . . . . . 14
⊢ (V ∩
𝐵) = 𝐵 |
21 | 20 | xpeq2i 5136 |
. . . . . . . . . . . . 13
⊢ (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵) |
22 | 17, 21 | eqtri 2644 |
. . . . . . . . . . . 12
⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵) |
23 | | snssi 4339 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴) |
24 | | df-ss 3588 |
. . . . . . . . . . . . . 14
⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶}) |
25 | 23, 24 | sylib 208 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ 𝐴 → ({𝐶} ∩ 𝐴) = {𝐶}) |
26 | 25 | xpeq1d 5138 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ 𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵)) |
27 | 22, 26 | syl5eq 2668 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵)) |
28 | 27 | reseq2d 5396 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵))) |
29 | 28 | rneqd 5353 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵))) |
30 | | 2ndconst 7266 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵) |
31 | | f1ofo 6144 |
. . . . . . . . . 10
⊢
((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵 → (2nd ↾
({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵) |
32 | | forn 6118 |
. . . . . . . . . 10
⊢
((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵) |
33 | 30, 31, 32 | 3syl 18 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵) |
34 | 29, 33 | eqtrd 2656 |
. . . . . . . 8
⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵) |
35 | 16, 34 | syl5eq 2668 |
. . . . . . 7
⊢ (𝐶 ∈ 𝐴 → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵) |
36 | 35 | adantl 482 |
. . . . . 6
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵) |
37 | 11, 36 | eqtrd 2656 |
. . . . 5
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵) |
38 | 8, 37 | syl5eq 2668 |
. . . 4
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵) |
39 | | curry1.1 |
. . . . . 6
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) |
40 | 39 | fneq1i 5985 |
. . . . 5
⊢ (𝐺 Fn 𝐵 ↔ (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵) |
41 | | df-fn 5891 |
. . . . 5
⊢ ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵)) |
42 | 40, 41 | bitri 264 |
. . . 4
⊢ (𝐺 Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵)) |
43 | 7, 38, 42 | sylanbrc 698 |
. . 3
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 Fn 𝐵) |
44 | | dffn5 6241 |
. . 3
⊢ (𝐺 Fn 𝐵 ↔ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥))) |
45 | 43, 44 | sylib 208 |
. 2
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥))) |
46 | 39 | fveq1i 6192 |
. . . . 5
⊢ (𝐺‘𝑥) = ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) |
47 | | dff1o4 6145 |
. . . . . . . . 9
⊢
((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
↔ ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V)) |
48 | 2, 47 | sylib 208 |
. . . . . . . 8
⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V)) |
49 | 48 | simprd 479 |
. . . . . . 7
⊢ (𝐶 ∈ 𝐴 → ◡(2nd ↾ ({𝐶} × V)) Fn V) |
50 | | vex 3203 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
51 | | fvco2 6273 |
. . . . . . . 8
⊢ ((◡(2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
52 | 50, 51 | mpan2 707 |
. . . . . . 7
⊢ (◡(2nd ↾ ({𝐶} × V)) Fn V → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
53 | 49, 52 | syl 17 |
. . . . . 6
⊢ (𝐶 ∈ 𝐴 → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
54 | 53 | ad2antlr 763 |
. . . . 5
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
55 | 46, 54 | syl5eq 2668 |
. . . 4
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥))) |
56 | 2 | adantr 481 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V) |
57 | | snidg 4206 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶}) |
58 | 57, 50 | jctir 561 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐴 → (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V)) |
59 | | opelxp 5146 |
. . . . . . . . . . 11
⊢
(〈𝐶, 𝑥〉 ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V)) |
60 | 58, 59 | sylibr 224 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → 〈𝐶, 𝑥〉 ∈ ({𝐶} × V)) |
61 | 60 | adantr 481 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 〈𝐶, 𝑥〉 ∈ ({𝐶} × V)) |
62 | 56, 61 | jca 554 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
∧ 〈𝐶, 𝑥〉 ∈ ({𝐶} × V))) |
63 | | fvres 6207 |
. . . . . . . . . . 11
⊢
(〈𝐶, 𝑥〉 ∈ ({𝐶} × V) →
((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = (2nd ‘〈𝐶, 𝑥〉)) |
64 | 60, 63 | syl 17 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = (2nd ‘〈𝐶, 𝑥〉)) |
65 | | op2ndg 7181 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ V) → (2nd
‘〈𝐶, 𝑥〉) = 𝑥) |
66 | 50, 65 | mpan2 707 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → (2nd ‘〈𝐶, 𝑥〉) = 𝑥) |
67 | 64, 66 | eqtrd 2656 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = 𝑥) |
68 | 67 | adantr 481 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = 𝑥) |
69 | | f1ocnvfv 6534 |
. . . . . . . 8
⊢
(((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V
∧ 〈𝐶, 𝑥〉 ∈ ({𝐶} × V)) →
(((2nd ↾ ({𝐶} × V))‘〈𝐶, 𝑥〉) = 𝑥 → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = 〈𝐶, 𝑥〉)) |
70 | 62, 68, 69 | sylc 65 |
. . . . . . 7
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = 〈𝐶, 𝑥〉) |
71 | 70 | fveq2d 6195 |
. . . . . 6
⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘〈𝐶, 𝑥〉)) |
72 | 71 | adantll 750 |
. . . . 5
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘〈𝐶, 𝑥〉)) |
73 | | df-ov 6653 |
. . . . 5
⊢ (𝐶𝐹𝑥) = (𝐹‘〈𝐶, 𝑥〉) |
74 | 72, 73 | syl6eqr 2674 |
. . . 4
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥)) |
75 | 55, 74 | eqtrd 2656 |
. . 3
⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐶𝐹𝑥)) |
76 | 75 | mpteq2dva 4744 |
. 2
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
77 | 45, 76 | eqtrd 2656 |
1
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |