Step | Hyp | Ref
| Expression |
1 | | f1od2.2 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊) |
2 | 1 | ralrimivva 2539 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊) |
3 | | f1od2.1 |
. . . 4
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
4 | 3 | fnmpo 6151 |
. . 3
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊 → 𝐹 Fn (𝐴 × 𝐵)) |
5 | 2, 4 | syl 14 |
. 2
⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵)) |
6 | | f1od2.3 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌)) |
7 | | opelxpi 4619 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌) → 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
8 | 6, 7 | syl 14 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
9 | 8 | ralrimiva 2530 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ 𝐷 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
10 | | eqid 2157 |
. . . . 5
⊢ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) = (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) |
11 | 10 | fnmpt 5297 |
. . . 4
⊢
(∀𝑧 ∈
𝐷 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌) → (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) Fn 𝐷) |
12 | 9, 11 | syl 14 |
. . 3
⊢ (𝜑 → (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) Fn 𝐷) |
13 | | elxp7 6119 |
. . . . . . . 8
⊢ (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵))) |
14 | 13 | anbi1i 454 |
. . . . . . 7
⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
15 | | anass 399 |
. . . . . . . . 9
⊢ (((𝑎 ∈ (V × V) ∧
((1st ‘𝑎)
∈ 𝐴 ∧
(2nd ‘𝑎)
∈ 𝐵)) ∧ 𝑧 =
⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑎 ∈ (V × V) ∧ (((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶))) |
16 | | f1od2.4 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) |
17 | 16 | sbcbidv 2995 |
. . . . . . . . . . . 12
⊢ (𝜑 → ([(2nd
‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(2nd
‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) |
18 | 17 | sbcbidv 2995 |
. . . . . . . . . . 11
⊢ (𝜑 → ([(1st
‘𝑎) / 𝑥][(2nd
‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st
‘𝑎) / 𝑥][(2nd
‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) |
19 | | sbcan 2979 |
. . . . . . . . . . . . . 14
⊢
([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ([(2nd
‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶)) |
20 | | sbcan 2979 |
. . . . . . . . . . . . . . . 16
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ ([(2nd
‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵)) |
21 | | vex 2715 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑎 ∈ V |
22 | | 2ndexg 6117 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ∈ V → (2nd
‘𝑎) ∈
V) |
23 | 21, 22 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘𝑎) ∈ V |
24 | | sbcg 3006 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
25 | 23, 24 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
26 | | sbcel1v 2999 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵) |
27 | 25, 26 | anbi12i 456 |
. . . . . . . . . . . . . . . 16
⊢
(([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
28 | 20, 27 | bitri 183 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
29 | | sbceq2g 3053 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
30 | 23, 29 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
31 | 28, 30 | anbi12i 456 |
. . . . . . . . . . . . . 14
⊢
(([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
32 | 19, 31 | bitri 183 |
. . . . . . . . . . . . 13
⊢
([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
33 | 32 | sbcbii 2996 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st
‘𝑎) / 𝑥]((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
34 | | sbcan 2979 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥]((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ ([(1st
‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ [(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
35 | | sbcan 2979 |
. . . . . . . . . . . . . 14
⊢
([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ([(1st
‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵)) |
36 | | sbcel1v 2999 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ↔ (1st ‘𝑎) ∈ 𝐴) |
37 | | 1stexg 6116 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ∈ V → (1st
‘𝑎) ∈
V) |
38 | 21, 37 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(1st ‘𝑎) ∈ V |
39 | | sbcg 3006 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥](2nd
‘𝑎) ∈ 𝐵 ↔ (2nd
‘𝑎) ∈ 𝐵)) |
40 | 38, 39 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵) |
41 | 36, 40 | anbi12i 456 |
. . . . . . . . . . . . . 14
⊢
(([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
42 | 35, 41 | bitri 183 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
43 | | sbceq2g 3053 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
44 | 38, 43 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
45 | 42, 44 | anbi12i 456 |
. . . . . . . . . . . 12
⊢
(([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ [(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
46 | 33, 34, 45 | 3bitri 205 |
. . . . . . . . . . 11
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
47 | | sbcan 2979 |
. . . . . . . . . . . . . 14
⊢
([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ ([(2nd
‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽))) |
48 | | sbcg 3006 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)) |
49 | 23, 48 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷) |
50 | | sbcan 2979 |
. . . . . . . . . . . . . . . 16
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ ([(2nd
‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽)) |
51 | | sbcg 3006 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼)) |
52 | 23, 51 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼) |
53 | | sbceq1g 3051 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2nd
‘𝑎) / 𝑦⦌𝑦 = 𝐽)) |
54 | 23, 53 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2nd
‘𝑎) / 𝑦⦌𝑦 = 𝐽) |
55 | | csbvarg 3059 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑎) ∈ V →
⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = (2nd ‘𝑎)) |
56 | 23, 55 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢
⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = (2nd ‘𝑎) |
57 | 56 | eqeq1i 2165 |
. . . . . . . . . . . . . . . . . 18
⊢
(⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
58 | 54, 57 | bitri 183 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
59 | 52, 58 | anbi12i 456 |
. . . . . . . . . . . . . . . 16
⊢
(([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
60 | 50, 59 | bitri 183 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
61 | 49, 60 | anbi12i 456 |
. . . . . . . . . . . . . 14
⊢
(([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
62 | 47, 61 | bitri 183 |
. . . . . . . . . . . . 13
⊢
([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
63 | 62 | sbcbii 2996 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ [(1st
‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
64 | | sbcan 2979 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ ([(1st
‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
65 | | sbcg 3006 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)) |
66 | 38, 65 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷) |
67 | | sbcan 2979 |
. . . . . . . . . . . . . 14
⊢
([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ([(1st
‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽)) |
68 | | sbceq1g 3051 |
. . . . . . . . . . . . . . . . 17
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st
‘𝑎) / 𝑥⦌𝑥 = 𝐼)) |
69 | 38, 68 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st
‘𝑎) / 𝑥⦌𝑥 = 𝐼) |
70 | | csbvarg 3059 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑎) ∈ V →
⦋(1st ‘𝑎) / 𝑥⦌𝑥 = (1st ‘𝑎)) |
71 | 38, 70 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
⦋(1st ‘𝑎) / 𝑥⦌𝑥 = (1st ‘𝑎) |
72 | 71 | eqeq1i 2165 |
. . . . . . . . . . . . . . . 16
⊢
(⦋(1st ‘𝑎) / 𝑥⦌𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼) |
73 | 69, 72 | bitri 183 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼) |
74 | | sbcg 3006 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥](2nd
‘𝑎) = 𝐽 ↔ (2nd
‘𝑎) = 𝐽)) |
75 | 38, 74 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
76 | 73, 75 | anbi12i 456 |
. . . . . . . . . . . . . 14
⊢
(([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
77 | 67, 76 | bitri 183 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
78 | 66, 77 | anbi12i 456 |
. . . . . . . . . . . 12
⊢
(([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
79 | 63, 64, 78 | 3bitri 205 |
. . . . . . . . . . 11
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
80 | 18, 46, 79 | 3bitr3g 221 |
. . . . . . . . . 10
⊢ (𝜑 → ((((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
81 | 80 | anbi2d 460 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))))) |
82 | 15, 81 | syl5bb 191 |
. . . . . . . 8
⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))))) |
83 | | xpss 4695 |
. . . . . . . . . . . 12
⊢ (𝑋 × 𝑌) ⊆ (V × V) |
84 | | simprr 522 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 = 〈𝐼, 𝐽〉) |
85 | 8 | adantrr 471 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
86 | 84, 85 | eqeltrd 2234 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 ∈ (𝑋 × 𝑌)) |
87 | 83, 86 | sseldi 3126 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 ∈ (V × V)) |
88 | 87 | ex 114 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) → 𝑎 ∈ (V × V))) |
89 | 88 | pm4.71rd 392 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)))) |
90 | | eqop 6126 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ (V × V) →
(𝑎 = 〈𝐼, 𝐽〉 ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
91 | 90 | anbi2d 460 |
. . . . . . . . . 10
⊢ (𝑎 ∈ (V × V) →
((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
92 | 91 | pm5.32i 450 |
. . . . . . . . 9
⊢ ((𝑎 ∈ (V × V) ∧
(𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
93 | 89, 92 | bitr2di 196 |
. . . . . . . 8
⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
94 | 82, 93 | bitrd 187 |
. . . . . . 7
⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
95 | 14, 94 | syl5bb 191 |
. . . . . 6
⊢ (𝜑 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
96 | 95 | opabbidv 4031 |
. . . . 5
⊢ (𝜑 → {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} = {〈𝑧, 𝑎〉 ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)}) |
97 | | df-mpo 5830 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
98 | 3, 97 | eqtri 2178 |
. . . . . . 7
⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
99 | 98 | cnveqi 4762 |
. . . . . 6
⊢ ◡𝐹 = ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
100 | | nfv 1508 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑎 ∈ (𝐴 × 𝐵) |
101 | | nfcsb1v 3064 |
. . . . . . . . 9
⊢
Ⅎ𝑥⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
102 | 101 | nfeq2 2311 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑧 =
⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
103 | 100, 102 | nfan 1545 |
. . . . . . 7
⊢
Ⅎ𝑥(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
104 | | nfv 1508 |
. . . . . . . 8
⊢
Ⅎ𝑦 𝑎 ∈ (𝐴 × 𝐵) |
105 | | nfcv 2299 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(1st ‘𝑎) |
106 | | nfcsb1v 3064 |
. . . . . . . . . 10
⊢
Ⅎ𝑦⦋(2nd ‘𝑎) / 𝑦⦌𝐶 |
107 | 105, 106 | nfcsb 3068 |
. . . . . . . . 9
⊢
Ⅎ𝑦⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
108 | 107 | nfeq2 2311 |
. . . . . . . 8
⊢
Ⅎ𝑦 𝑧 =
⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
109 | 104, 108 | nfan 1545 |
. . . . . . 7
⊢
Ⅎ𝑦(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
110 | | eleq1 2220 |
. . . . . . . . 9
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝑎 ∈ (𝐴 × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) |
111 | | opelxp 4617 |
. . . . . . . . 9
⊢
(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
112 | 110, 111 | bitrdi 195 |
. . . . . . . 8
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
113 | | csbopeq1a 6137 |
. . . . . . . . 9
⊢ (𝑎 = 〈𝑥, 𝑦〉 → ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 = 𝐶) |
114 | 113 | eqeq2d 2169 |
. . . . . . . 8
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = 𝐶)) |
115 | 112, 114 | anbi12d 465 |
. . . . . . 7
⊢ (𝑎 = 〈𝑥, 𝑦〉 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))) |
116 | | xpss 4695 |
. . . . . . . . 9
⊢ (𝐴 × 𝐵) ⊆ (V × V) |
117 | 116 | sseli 3124 |
. . . . . . . 8
⊢ (𝑎 ∈ (𝐴 × 𝐵) → 𝑎 ∈ (V × V)) |
118 | 117 | adantr 274 |
. . . . . . 7
⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) → 𝑎 ∈ (V × V)) |
119 | 103, 109,
115, 118 | cnvoprab 6182 |
. . . . . 6
⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} |
120 | 99, 119 | eqtri 2178 |
. . . . 5
⊢ ◡𝐹 = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} |
121 | | df-mpt 4028 |
. . . . 5
⊢ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) = {〈𝑧, 𝑎〉 ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)} |
122 | 96, 120, 121 | 3eqtr4g 2215 |
. . . 4
⊢ (𝜑 → ◡𝐹 = (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉)) |
123 | 122 | fneq1d 5261 |
. . 3
⊢ (𝜑 → (◡𝐹 Fn 𝐷 ↔ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) Fn 𝐷)) |
124 | 12, 123 | mpbird 166 |
. 2
⊢ (𝜑 → ◡𝐹 Fn 𝐷) |
125 | | dff1o4 5423 |
. 2
⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ◡𝐹 Fn 𝐷)) |
126 | 5, 124, 125 | sylanbrc 414 |
1
⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷) |