Step | Hyp | Ref
| Expression |
1 | | f1od2.2 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊) |
2 | 1 | ralrimivva 3115 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊) |
3 | | f1od2.1 |
. . . 4
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
4 | 3 | fnmpo 7898 |
. . 3
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊 → 𝐹 Fn (𝐴 × 𝐵)) |
5 | 2, 4 | syl 17 |
. 2
⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵)) |
6 | | f1od2.3 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌)) |
7 | | opelxpi 5621 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌) → 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
8 | 6, 7 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
9 | 8 | ralrimiva 3108 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ 𝐷 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
10 | | eqid 2738 |
. . . . 5
⊢ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) = (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) |
11 | 10 | fnmpt 6565 |
. . . 4
⊢
(∀𝑧 ∈
𝐷 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌) → (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) Fn 𝐷) |
12 | 9, 11 | syl 17 |
. . 3
⊢ (𝜑 → (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) Fn 𝐷) |
13 | | elxp7 7855 |
. . . . . . . 8
⊢ (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵))) |
14 | 13 | anbi1i 624 |
. . . . . . 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 469 |
. . . . . . . . 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 3774 |
. . . . . . . . . . . 12
⊢ (𝜑 → ([(2nd
‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(2nd
‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) |
18 | 17 | sbcbidv 3774 |
. . . . . . . . . . 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 3767 |
. . . . . . . . . . . . . 14
⊢
([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ([(2nd
‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶)) |
20 | | sbcan 3767 |
. . . . . . . . . . . . . . . 16
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ ([(2nd
‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵)) |
21 | | fvex 6779 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘𝑎) ∈ V |
22 | | sbcg 3794 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
23 | 21, 22 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
24 | | sbcel1v 3786 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵) |
25 | 23, 24 | anbi12i 627 |
. . . . . . . . . . . . . . . 16
⊢
(([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
26 | 20, 25 | bitri 274 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
27 | | sbceq2g 4350 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
28 | 21, 27 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
29 | 26, 28 | anbi12i 627 |
. . . . . . . . . . . . . 14
⊢
(([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
30 | 19, 29 | bitri 274 |
. . . . . . . . . . . . 13
⊢
([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
31 | 30 | sbcbii 3775 |
. . . . . . . . . . . 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)) |
32 | | sbcan 3767 |
. . . . . . . . . . . 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)) |
33 | | sbcan 3767 |
. . . . . . . . . . . . . 14
⊢
([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ([(1st
‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵)) |
34 | | sbcel1v 3786 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ↔ (1st ‘𝑎) ∈ 𝐴) |
35 | | fvex 6779 |
. . . . . . . . . . . . . . . 16
⊢
(1st ‘𝑎) ∈ V |
36 | | sbcg 3794 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥](2nd
‘𝑎) ∈ 𝐵 ↔ (2nd
‘𝑎) ∈ 𝐵)) |
37 | 35, 36 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵) |
38 | 34, 37 | anbi12i 627 |
. . . . . . . . . . . . . 14
⊢
(([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
39 | 33, 38 | bitri 274 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
40 | | sbceq2g 4350 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
41 | 35, 40 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
42 | 39, 41 | anbi12i 627 |
. . . . . . . . . . . 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)) |
43 | 31, 32, 42 | 3bitri 297 |
. . . . . . . . . . 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)) |
44 | | sbcan 3767 |
. . . . . . . . . . . . . 14
⊢
([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ ([(2nd
‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽))) |
45 | | sbcg 3794 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)) |
46 | 21, 45 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷) |
47 | | sbcan 3767 |
. . . . . . . . . . . . . . . 16
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ ([(2nd
‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽)) |
48 | | sbcg 3794 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼)) |
49 | 21, 48 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼) |
50 | | sbceq1g 4348 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑎) ∈ V → ([(2nd
‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2nd
‘𝑎) / 𝑦⦌𝑦 = 𝐽)) |
51 | 21, 50 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2nd
‘𝑎) / 𝑦⦌𝑦 = 𝐽) |
52 | 21 | csbvargi 4366 |
. . . . . . . . . . . . . . . . . . 19
⊢
⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = (2nd ‘𝑎) |
53 | 52 | eqeq1i 2743 |
. . . . . . . . . . . . . . . . . 18
⊢
(⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
54 | 51, 53 | bitri 274 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
55 | 49, 54 | anbi12i 627 |
. . . . . . . . . . . . . . . 16
⊢
(([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
56 | 47, 55 | bitri 274 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
57 | 46, 56 | anbi12i 627 |
. . . . . . . . . . . . . 14
⊢
(([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
58 | 44, 57 | bitri 274 |
. . . . . . . . . . . . 13
⊢
([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
59 | 58 | sbcbii 3775 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ [(1st
‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
60 | | sbcan 3767 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ ([(1st
‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
61 | | sbcg 3794 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)) |
62 | 35, 61 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷) |
63 | | sbcan 3767 |
. . . . . . . . . . . . . 14
⊢
([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ([(1st
‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽)) |
64 | | sbceq1g 4348 |
. . . . . . . . . . . . . . . . 17
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st
‘𝑎) / 𝑥⦌𝑥 = 𝐼)) |
65 | 35, 64 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st
‘𝑎) / 𝑥⦌𝑥 = 𝐼) |
66 | 35 | csbvargi 4366 |
. . . . . . . . . . . . . . . . 17
⊢
⦋(1st ‘𝑎) / 𝑥⦌𝑥 = (1st ‘𝑎) |
67 | 66 | eqeq1i 2743 |
. . . . . . . . . . . . . . . 16
⊢
(⦋(1st ‘𝑎) / 𝑥⦌𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼) |
68 | 65, 67 | bitri 274 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼) |
69 | | sbcg 3794 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥](2nd
‘𝑎) = 𝐽 ↔ (2nd
‘𝑎) = 𝐽)) |
70 | 35, 69 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
71 | 68, 70 | anbi12i 627 |
. . . . . . . . . . . . . 14
⊢
(([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
72 | 63, 71 | bitri 274 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
73 | 62, 72 | anbi12i 627 |
. . . . . . . . . . . 12
⊢
(([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
74 | 59, 60, 73 | 3bitri 297 |
. . . . . . . . . . 11
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
75 | 18, 43, 74 | 3bitr3g 313 |
. . . . . . . . . 10
⊢ (𝜑 → ((((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
76 | 75 | anbi2d 629 |
. . . . . . . . 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))))) |
77 | 15, 76 | syl5bb 283 |
. . . . . . . 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))))) |
78 | | xpss 5600 |
. . . . . . . . . . . 12
⊢ (𝑋 × 𝑌) ⊆ (V × V) |
79 | | simprr 770 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 = 〈𝐼, 𝐽〉) |
80 | 8 | adantrr 714 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
81 | 79, 80 | eqeltrd 2839 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 ∈ (𝑋 × 𝑌)) |
82 | 78, 81 | sselid 3918 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 ∈ (V × V)) |
83 | 82 | ex 413 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) → 𝑎 ∈ (V × V))) |
84 | 83 | pm4.71rd 563 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)))) |
85 | | eqop 7862 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ (V × V) →
(𝑎 = 〈𝐼, 𝐽〉 ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
86 | 85 | anbi2d 629 |
. . . . . . . . . 10
⊢ (𝑎 ∈ (V × V) →
((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
87 | 86 | pm5.32i 575 |
. . . . . . . . 9
⊢ ((𝑎 ∈ (V × V) ∧
(𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
88 | 84, 87 | bitr2di 288 |
. . . . . . . 8
⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
89 | 77, 88 | bitrd 278 |
. . . . . . 7
⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
90 | 14, 89 | syl5bb 283 |
. . . . . 6
⊢ (𝜑 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
91 | 90 | opabbidv 5139 |
. . . . 5
⊢ (𝜑 → {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} = {〈𝑧, 𝑎〉 ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)}) |
92 | | df-mpo 7272 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
93 | 3, 92 | eqtri 2766 |
. . . . . . 7
⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
94 | 93 | cnveqi 5776 |
. . . . . 6
⊢ ◡𝐹 = ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
95 | | nfv 1917 |
. . . . . . . 8
⊢
Ⅎ𝑖((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) |
96 | | nfv 1917 |
. . . . . . . 8
⊢
Ⅎ𝑗((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) |
97 | | nfv 1917 |
. . . . . . . . 9
⊢
Ⅎ𝑥(𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) |
98 | | nfcsb1v 3856 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 |
99 | 98 | nfeq2 2924 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 |
100 | 97, 99 | nfan 1902 |
. . . . . . . 8
⊢
Ⅎ𝑥((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) ∧ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶) |
101 | | nfv 1917 |
. . . . . . . . 9
⊢
Ⅎ𝑦(𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) |
102 | | nfcv 2907 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦𝑖 |
103 | | nfcsb1v 3856 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦⦋𝑗 / 𝑦⦌𝐶 |
104 | 102, 103 | nfcsbw 3858 |
. . . . . . . . . 10
⊢
Ⅎ𝑦⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 |
105 | 104 | nfeq2 2924 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 |
106 | 101, 105 | nfan 1902 |
. . . . . . . 8
⊢
Ⅎ𝑦((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) ∧ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶) |
107 | | simpl 483 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → 𝑥 = 𝑖) |
108 | 107 | eleq1d 2823 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)) |
109 | | simpr 485 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → 𝑦 = 𝑗) |
110 | 109 | eleq1d 2823 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑦 ∈ 𝐵 ↔ 𝑗 ∈ 𝐵)) |
111 | 108, 110 | anbi12d 631 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵))) |
112 | | csbeq1a 3845 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑗 → 𝐶 = ⦋𝑗 / 𝑦⦌𝐶) |
113 | | csbeq1a 3845 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑖 → ⦋𝑗 / 𝑦⦌𝐶 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶) |
114 | 112, 113 | sylan9eqr 2800 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → 𝐶 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶) |
115 | 114 | eqeq2d 2749 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑧 = 𝐶 ↔ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶)) |
116 | 111, 115 | anbi12d 631 |
. . . . . . . 8
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) ∧ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶))) |
117 | 95, 96, 100, 106, 116 | cbvoprab12 7354 |
. . . . . . 7
⊢
{〈〈𝑥,
𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {〈〈𝑖, 𝑗〉, 𝑧〉 ∣ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) ∧ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶)} |
118 | 117 | cnveqi 5776 |
. . . . . 6
⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = ◡{〈〈𝑖, 𝑗〉, 𝑧〉 ∣ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) ∧ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶)} |
119 | | eleq1 2826 |
. . . . . . . . 9
⊢ (𝑎 = 〈𝑖, 𝑗〉 → (𝑎 ∈ (𝐴 × 𝐵) ↔ 〈𝑖, 𝑗〉 ∈ (𝐴 × 𝐵))) |
120 | | opelxp 5620 |
. . . . . . . . 9
⊢
(〈𝑖, 𝑗〉 ∈ (𝐴 × 𝐵) ↔ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) |
121 | 119, 120 | bitrdi 287 |
. . . . . . . 8
⊢ (𝑎 = 〈𝑖, 𝑗〉 → (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵))) |
122 | | csbcom 4351 |
. . . . . . . . . . . . 13
⊢
⦋(2nd ‘𝑎) / 𝑗⦌⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 = ⦋𝑖 / 𝑥⦌⦋(2nd
‘𝑎) / 𝑗⦌⦋𝑗 / 𝑦⦌𝐶 |
123 | | csbcow 3846 |
. . . . . . . . . . . . . 14
⊢
⦋(2nd ‘𝑎) / 𝑗⦌⦋𝑗 / 𝑦⦌𝐶 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
124 | 123 | csbeq2i 3839 |
. . . . . . . . . . . . 13
⊢
⦋𝑖 /
𝑥⦌⦋(2nd
‘𝑎) / 𝑗⦌⦋𝑗 / 𝑦⦌𝐶 = ⦋𝑖 / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
125 | 122, 124 | eqtri 2766 |
. . . . . . . . . . . 12
⊢
⦋(2nd ‘𝑎) / 𝑗⦌⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 = ⦋𝑖 / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
126 | 125 | csbeq2i 3839 |
. . . . . . . . . . 11
⊢
⦋(1st ‘𝑎) / 𝑖⦌⦋(2nd
‘𝑎) / 𝑗⦌⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 = ⦋(1st
‘𝑎) / 𝑖⦌⦋𝑖 / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
127 | | csbcow 3846 |
. . . . . . . . . . 11
⊢
⦋(1st ‘𝑎) / 𝑖⦌⦋𝑖 / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
128 | 126, 127 | eqtri 2766 |
. . . . . . . . . 10
⊢
⦋(1st ‘𝑎) / 𝑖⦌⦋(2nd
‘𝑎) / 𝑗⦌⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
129 | | csbopeq1a 7880 |
. . . . . . . . . 10
⊢ (𝑎 = 〈𝑖, 𝑗〉 → ⦋(1st
‘𝑎) / 𝑖⦌⦋(2nd
‘𝑎) / 𝑗⦌⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶) |
130 | 128, 129 | eqtr3id 2792 |
. . . . . . . . 9
⊢ (𝑎 = 〈𝑖, 𝑗〉 → ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶) |
131 | 130 | eqeq2d 2749 |
. . . . . . . 8
⊢ (𝑎 = 〈𝑖, 𝑗〉 → (𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶)) |
132 | 121, 131 | anbi12d 631 |
. . . . . . 7
⊢ (𝑎 = 〈𝑖, 𝑗〉 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) ∧ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶))) |
133 | | xpss 5600 |
. . . . . . . . 9
⊢ (𝐴 × 𝐵) ⊆ (V × V) |
134 | 133 | sseli 3916 |
. . . . . . . 8
⊢ (𝑎 ∈ (𝐴 × 𝐵) → 𝑎 ∈ (V × V)) |
135 | 134 | adantr 481 |
. . . . . . 7
⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) → 𝑎 ∈ (V × V)) |
136 | 132, 135 | cnvoprab 7889 |
. . . . . 6
⊢ ◡{〈〈𝑖, 𝑗〉, 𝑧〉 ∣ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) ∧ 𝑧 = ⦋𝑖 / 𝑥⦌⦋𝑗 / 𝑦⦌𝐶)} = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} |
137 | 94, 118, 136 | 3eqtri 2770 |
. . . . 5
⊢ ◡𝐹 = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} |
138 | | df-mpt 5157 |
. . . . 5
⊢ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) = {〈𝑧, 𝑎〉 ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)} |
139 | 91, 137, 138 | 3eqtr4g 2803 |
. . . 4
⊢ (𝜑 → ◡𝐹 = (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉)) |
140 | 139 | fneq1d 6518 |
. . 3
⊢ (𝜑 → (◡𝐹 Fn 𝐷 ↔ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) Fn 𝐷)) |
141 | 12, 140 | mpbird 256 |
. 2
⊢ (𝜑 → ◡𝐹 Fn 𝐷) |
142 | | dff1o4 6716 |
. 2
⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ◡𝐹 Fn 𝐷)) |
143 | 5, 141, 142 | sylanbrc 583 |
1
⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷) |