Step | Hyp | Ref
| Expression |
1 | | f1od2.2 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊) |
2 | 1 | ralrimivva 2954 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊) |
3 | | f1od2.1 |
. . . 4
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
4 | 3 | fnmpt2 7127 |
. . 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 5072 |
. . . . . 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 2949 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ 𝐷 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
10 | | eqid 2610 |
. . . . 5
⊢ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) = (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) |
11 | 10 | fnmpt 5933 |
. . . 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 7092 |
. . . . . . . 8
⊢ (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵))) |
14 | 13 | anbi1i 727 |
. . . . . . 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 679 |
. . . . . . . . 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 3457 |
. . . . . . . . . . . 12
⊢ (𝜑 → ([(2nd
‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(2nd
‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) |
18 | 17 | sbcbidv 3457 |
. . . . . . . . . . 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 3445 |
. . . . . . . . . . . . . 14
⊢
([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ([(2nd
‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶)) |
20 | | sbcan 3445 |
. . . . . . . . . . . . . . . 16
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ ([(2nd
‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵)) |
21 | | fvex 6113 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘𝑎) ∈ V |
22 | | sbcg 3470 |
. . . . . . . . . . . . . . . . . 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 3462 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵) |
25 | 23, 24 | anbi12i 729 |
. . . . . . . . . . . . . . . 16
⊢
(([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
26 | 20, 25 | bitri 263 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
27 | | sbceq2g 3942 |
. . . . . . . . . . . . . . . 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 729 |
. . . . . . . . . . . . . 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 263 |
. . . . . . . . . . . . 13
⊢
([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd
‘𝑎) / 𝑦⦌𝐶)) |
31 | 30 | sbcbii 3458 |
. . . . . . . . . . . 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 3445 |
. . . . . . . . . . . 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 3445 |
. . . . . . . . . . . . . 14
⊢
([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ([(1st
‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵)) |
34 | | sbcel1v 3462 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ↔ (1st ‘𝑎) ∈ 𝐴) |
35 | | fvex 6113 |
. . . . . . . . . . . . . . . 16
⊢
(1st ‘𝑎) ∈ V |
36 | | sbcg 3470 |
. . . . . . . . . . . . . . . 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 729 |
. . . . . . . . . . . . . 14
⊢
(([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
39 | 33, 38 | bitri 263 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) |
40 | | sbceq2g 3942 |
. . . . . . . . . . . . . 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 729 |
. . . . . . . . . . . 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 285 |
. . . . . . . . . . 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 3445 |
. . . . . . . . . . . . . 14
⊢
([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ ([(2nd
‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽))) |
45 | | sbcg 3470 |
. . . . . . . . . . . . . . . 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 3445 |
. . . . . . . . . . . . . . . 16
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ ([(2nd
‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽)) |
48 | | sbcg 3470 |
. . . . . . . . . . . . . . . . . 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 3940 |
. . . . . . . . . . . . . . . . . . 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 | | csbvarg 3955 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑎) ∈ V →
⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = (2nd ‘𝑎)) |
53 | 21, 52 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢
⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = (2nd ‘𝑎) |
54 | 53 | eqeq1i 2615 |
. . . . . . . . . . . . . . . . . 18
⊢
(⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
55 | 51, 54 | bitri 263 |
. . . . . . . . . . . . . . . . 17
⊢
([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
56 | 49, 55 | anbi12i 729 |
. . . . . . . . . . . . . . . 16
⊢
(([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
57 | 47, 56 | bitri 263 |
. . . . . . . . . . . . . . 15
⊢
([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
58 | 46, 57 | anbi12i 729 |
. . . . . . . . . . . . . 14
⊢
(([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
59 | 44, 58 | bitri 263 |
. . . . . . . . . . . . 13
⊢
([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
60 | 59 | sbcbii 3458 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ [(1st
‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
61 | | sbcan 3445 |
. . . . . . . . . . . 12
⊢
([(1st ‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ ([(1st
‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
62 | | sbcg 3470 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)) |
63 | 35, 62 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷) |
64 | | sbcan 3445 |
. . . . . . . . . . . . . 14
⊢
([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ([(1st
‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽)) |
65 | | sbceq1g 3940 |
. . . . . . . . . . . . . . . . 17
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st
‘𝑎) / 𝑥⦌𝑥 = 𝐼)) |
66 | 35, 65 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st
‘𝑎) / 𝑥⦌𝑥 = 𝐼) |
67 | | csbvarg 3955 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑎) ∈ V →
⦋(1st ‘𝑎) / 𝑥⦌𝑥 = (1st ‘𝑎)) |
68 | 35, 67 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
⦋(1st ‘𝑎) / 𝑥⦌𝑥 = (1st ‘𝑎) |
69 | 68 | eqeq1i 2615 |
. . . . . . . . . . . . . . . 16
⊢
(⦋(1st ‘𝑎) / 𝑥⦌𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼) |
70 | 66, 69 | bitri 263 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼) |
71 | | sbcg 3470 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑎) ∈ V → ([(1st
‘𝑎) / 𝑥](2nd
‘𝑎) = 𝐽 ↔ (2nd
‘𝑎) = 𝐽)) |
72 | 35, 71 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽 ↔ (2nd ‘𝑎) = 𝐽) |
73 | 70, 72 | anbi12i 729 |
. . . . . . . . . . . . . 14
⊢
(([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
74 | 64, 73 | bitri 263 |
. . . . . . . . . . . . 13
⊢
([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) |
75 | 63, 74 | anbi12i 729 |
. . . . . . . . . . . 12
⊢
(([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
76 | 60, 61, 75 | 3bitri 285 |
. . . . . . . . . . 11
⊢
([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
77 | 18, 43, 76 | 3bitr3g 301 |
. . . . . . . . . 10
⊢ (𝜑 → ((((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
78 | 77 | anbi2d 736 |
. . . . . . . . 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))))) |
79 | 15, 78 | syl5bb 271 |
. . . . . . . 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))))) |
80 | | xpss 5149 |
. . . . . . . . . . . 12
⊢ (𝑋 × 𝑌) ⊆ (V × V) |
81 | | simprr 792 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 = 〈𝐼, 𝐽〉) |
82 | 8 | adantrr 749 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 〈𝐼, 𝐽〉 ∈ (𝑋 × 𝑌)) |
83 | 81, 82 | eqeltrd 2688 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 ∈ (𝑋 × 𝑌)) |
84 | 80, 83 | sseldi 3566 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) → 𝑎 ∈ (V × V)) |
85 | 84 | ex 449 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) → 𝑎 ∈ (V × V))) |
86 | 85 | pm4.71rd 665 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)))) |
87 | | eqop 7099 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ (V × V) →
(𝑎 = 〈𝐼, 𝐽〉 ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) |
88 | 87 | anbi2d 736 |
. . . . . . . . . 10
⊢ (𝑎 ∈ (V × V) →
((𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
89 | 88 | pm5.32i 667 |
. . . . . . . . 9
⊢ ((𝑎 ∈ (V × V) ∧
(𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))) |
90 | 86, 89 | syl6rbb 276 |
. . . . . . . 8
⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
91 | 79, 90 | bitrd 267 |
. . . . . . 7
⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st
‘𝑎) ∈ 𝐴 ∧ (2nd
‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
92 | 14, 91 | syl5bb 271 |
. . . . . 6
⊢ (𝜑 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉))) |
93 | 92 | opabbidv 4648 |
. . . . 5
⊢ (𝜑 → {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} = {〈𝑧, 𝑎〉 ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)}) |
94 | | df-mpt2 6554 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
95 | 3, 94 | eqtri 2632 |
. . . . . . 7
⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
96 | 95 | cnveqi 5219 |
. . . . . 6
⊢ ◡𝐹 = ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
97 | | nfv 1830 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑎 ∈ (𝐴 × 𝐵) |
98 | | nfcsb1v 3515 |
. . . . . . . . 9
⊢
Ⅎ𝑥⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
99 | 98 | nfeq2 2766 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑧 =
⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
100 | 97, 99 | nfan 1816 |
. . . . . . 7
⊢
Ⅎ𝑥(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
101 | | nfv 1830 |
. . . . . . . 8
⊢
Ⅎ𝑦 𝑎 ∈ (𝐴 × 𝐵) |
102 | | nfcv 2751 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(1st ‘𝑎) |
103 | | nfcsb1v 3515 |
. . . . . . . . . 10
⊢
Ⅎ𝑦⦋(2nd ‘𝑎) / 𝑦⦌𝐶 |
104 | 102, 103 | nfcsb 3517 |
. . . . . . . . 9
⊢
Ⅎ𝑦⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
105 | 104 | nfeq2 2766 |
. . . . . . . 8
⊢
Ⅎ𝑦 𝑧 =
⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 |
106 | 101, 105 | nfan 1816 |
. . . . . . 7
⊢
Ⅎ𝑦(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) |
107 | | eleq1 2676 |
. . . . . . . . 9
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝑎 ∈ (𝐴 × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) |
108 | | opelxp 5070 |
. . . . . . . . 9
⊢
(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
109 | 107, 108 | syl6bb 275 |
. . . . . . . 8
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
110 | | csbopeq1a 7112 |
. . . . . . . . 9
⊢ (𝑎 = 〈𝑥, 𝑦〉 → ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 = 𝐶) |
111 | 110 | eqeq2d 2620 |
. . . . . . . 8
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = 𝐶)) |
112 | 109, 111 | anbi12d 743 |
. . . . . . 7
⊢ (𝑎 = 〈𝑥, 𝑦〉 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))) |
113 | | xpss 5149 |
. . . . . . . . 9
⊢ (𝐴 × 𝐵) ⊆ (V × V) |
114 | 113 | sseli 3564 |
. . . . . . . 8
⊢ (𝑎 ∈ (𝐴 × 𝐵) → 𝑎 ∈ (V × V)) |
115 | 114 | adantr 480 |
. . . . . . 7
⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶) → 𝑎 ∈ (V × V)) |
116 | 100, 106,
112, 115 | cnvoprab 28886 |
. . . . . 6
⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} |
117 | 96, 116 | eqtri 2632 |
. . . . 5
⊢ ◡𝐹 = {〈𝑧, 𝑎〉 ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st
‘𝑎) / 𝑥⦌⦋(2nd
‘𝑎) / 𝑦⦌𝐶)} |
118 | | df-mpt 4645 |
. . . . 5
⊢ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) = {〈𝑧, 𝑎〉 ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = 〈𝐼, 𝐽〉)} |
119 | 93, 117, 118 | 3eqtr4g 2669 |
. . . 4
⊢ (𝜑 → ◡𝐹 = (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉)) |
120 | 119 | fneq1d 5895 |
. . 3
⊢ (𝜑 → (◡𝐹 Fn 𝐷 ↔ (𝑧 ∈ 𝐷 ↦ 〈𝐼, 𝐽〉) Fn 𝐷)) |
121 | 12, 120 | mpbird 246 |
. 2
⊢ (𝜑 → ◡𝐹 Fn 𝐷) |
122 | | dff1o4 6058 |
. 2
⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ◡𝐹 Fn 𝐷)) |
123 | 5, 121, 122 | sylanbrc 695 |
1
⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷) |