Proof of Theorem fnwelem
Step | Hyp | Ref
| Expression |
1 | | fnwe.2 |
. . . 4
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
2 | | ffvelrn 6959 |
. . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
3 | | simpr 485 |
. . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴) |
4 | 2, 3 | opelxpd 5627 |
. . . . 5
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → 〈(𝐹‘𝑧), 𝑧〉 ∈ (𝐵 × 𝐴)) |
5 | | fnwelem.7 |
. . . . 5
⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) |
6 | 4, 5 | fmptd 6988 |
. . . 4
⊢ (𝐹:𝐴⟶𝐵 → 𝐺:𝐴⟶(𝐵 × 𝐴)) |
7 | | frn 6607 |
. . . 4
⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → ran 𝐺 ⊆ (𝐵 × 𝐴)) |
8 | 1, 6, 7 | 3syl 18 |
. . 3
⊢ (𝜑 → ran 𝐺 ⊆ (𝐵 × 𝐴)) |
9 | | fnwe.3 |
. . . 4
⊢ (𝜑 → 𝑅 We 𝐵) |
10 | | fnwe.4 |
. . . 4
⊢ (𝜑 → 𝑆 We 𝐴) |
11 | | fnwelem.6 |
. . . . 5
⊢ 𝑄 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd
‘𝑢)𝑆(2nd ‘𝑣))))} |
12 | 11 | wexp 7971 |
. . . 4
⊢ ((𝑅 We 𝐵 ∧ 𝑆 We 𝐴) → 𝑄 We (𝐵 × 𝐴)) |
13 | 9, 10, 12 | syl2anc 584 |
. . 3
⊢ (𝜑 → 𝑄 We (𝐵 × 𝐴)) |
14 | | wess 5576 |
. . 3
⊢ (ran
𝐺 ⊆ (𝐵 × 𝐴) → (𝑄 We (𝐵 × 𝐴) → 𝑄 We ran 𝐺)) |
15 | 8, 13, 14 | sylc 65 |
. 2
⊢ (𝜑 → 𝑄 We ran 𝐺) |
16 | | fveq2 6774 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) |
17 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥) |
18 | 16, 17 | opeq12d 4812 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑥 → 〈(𝐹‘𝑧), 𝑧〉 = 〈(𝐹‘𝑥), 𝑥〉) |
19 | | opex 5379 |
. . . . . . . . . . 11
⊢
〈(𝐹‘𝑥), 𝑥〉 ∈ V |
20 | 18, 5, 19 | fvmpt 6875 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝐴 → (𝐺‘𝑥) = 〈(𝐹‘𝑥), 𝑥〉) |
21 | | fveq2 6774 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦)) |
22 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) |
23 | 21, 22 | opeq12d 4812 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → 〈(𝐹‘𝑧), 𝑧〉 = 〈(𝐹‘𝑦), 𝑦〉) |
24 | | opex 5379 |
. . . . . . . . . . 11
⊢
〈(𝐹‘𝑦), 𝑦〉 ∈ V |
25 | 23, 5, 24 | fvmpt 6875 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝐴 → (𝐺‘𝑦) = 〈(𝐹‘𝑦), 𝑦〉) |
26 | 20, 25 | eqeqan12d 2752 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐺‘𝑦) ↔ 〈(𝐹‘𝑥), 𝑥〉 = 〈(𝐹‘𝑦), 𝑦〉)) |
27 | | fvex 6787 |
. . . . . . . . . . 11
⊢ (𝐹‘𝑥) ∈ V |
28 | | vex 3436 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
29 | 27, 28 | opth 5391 |
. . . . . . . . . 10
⊢
(〈(𝐹‘𝑥), 𝑥〉 = 〈(𝐹‘𝑦), 𝑦〉 ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 = 𝑦)) |
30 | 29 | simprbi 497 |
. . . . . . . . 9
⊢
(〈(𝐹‘𝑥), 𝑥〉 = 〈(𝐹‘𝑦), 𝑦〉 → 𝑥 = 𝑦) |
31 | 26, 30 | syl6bi 252 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦)) |
32 | 31 | rgen2 3120 |
. . . . . . 7
⊢
∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦) |
33 | | dff13 7128 |
. . . . . . 7
⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) ↔ (𝐺:𝐴⟶(𝐵 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))) |
34 | 6, 32, 33 | sylanblrc 590 |
. . . . . 6
⊢ (𝐹:𝐴⟶𝐵 → 𝐺:𝐴–1-1→(𝐵 × 𝐴)) |
35 | | f1f1orn 6727 |
. . . . . 6
⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) → 𝐺:𝐴–1-1-onto→ran
𝐺) |
36 | | f1ocnv 6728 |
. . . . . 6
⊢ (𝐺:𝐴–1-1-onto→ran
𝐺 → ◡𝐺:ran 𝐺–1-1-onto→𝐴) |
37 | 1, 34, 35, 36 | 4syl 19 |
. . . . 5
⊢ (𝜑 → ◡𝐺:ran 𝐺–1-1-onto→𝐴) |
38 | | eqid 2738 |
. . . . . . 7
⊢
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} |
39 | 38 | f1oiso2 7223 |
. . . . . 6
⊢ (◡𝐺:ran 𝐺–1-1-onto→𝐴 → ◡𝐺 Isom 𝑄, {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} (ran 𝐺, 𝐴)) |
40 | | fnwe.1 |
. . . . . . . 8
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} |
41 | | frel 6605 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → Rel 𝐺) |
42 | | dfrel2 6092 |
. . . . . . . . . . . . . . . 16
⊢ (Rel
𝐺 ↔ ◡◡𝐺 = 𝐺) |
43 | 41, 42 | sylib 217 |
. . . . . . . . . . . . . . 15
⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → ◡◡𝐺 = 𝐺) |
44 | 43 | fveq1d 6776 |
. . . . . . . . . . . . . 14
⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → (◡◡𝐺‘𝑥) = (𝐺‘𝑥)) |
45 | 43 | fveq1d 6776 |
. . . . . . . . . . . . . 14
⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → (◡◡𝐺‘𝑦) = (𝐺‘𝑦)) |
46 | 44, 45 | breq12d 5087 |
. . . . . . . . . . . . 13
⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → ((◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦) ↔ (𝐺‘𝑥)𝑄(𝐺‘𝑦))) |
47 | 6, 46 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐹:𝐴⟶𝐵 → ((◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦) ↔ (𝐺‘𝑥)𝑄(𝐺‘𝑦))) |
48 | 47 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦) ↔ (𝐺‘𝑥)𝑄(𝐺‘𝑦))) |
49 | 20, 25 | breqan12d 5090 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑥)𝑄(𝐺‘𝑦) ↔ 〈(𝐹‘𝑥), 𝑥〉𝑄〈(𝐹‘𝑦), 𝑦〉)) |
50 | 49 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐺‘𝑥)𝑄(𝐺‘𝑦) ↔ 〈(𝐹‘𝑥), 𝑥〉𝑄〈(𝐹‘𝑦), 𝑦〉)) |
51 | | eleq1 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → (𝑢 ∈ (𝐵 × 𝐴) ↔ 〈(𝐹‘𝑥), 𝑥〉 ∈ (𝐵 × 𝐴))) |
52 | | opelxp 5625 |
. . . . . . . . . . . . . . . 16
⊢
(〈(𝐹‘𝑥), 𝑥〉 ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) |
53 | 51, 52 | bitrdi 287 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → (𝑢 ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))) |
54 | 53 | anbi1d 630 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)))) |
55 | 27, 28 | op1std 7841 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → (1st ‘𝑢) = (𝐹‘𝑥)) |
56 | 55 | breq1d 5084 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → ((1st ‘𝑢)𝑅(1st ‘𝑣) ↔ (𝐹‘𝑥)𝑅(1st ‘𝑣))) |
57 | 55 | eqeq1d 2740 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → ((1st ‘𝑢) = (1st ‘𝑣) ↔ (𝐹‘𝑥) = (1st ‘𝑣))) |
58 | 27, 28 | op2ndd 7842 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → (2nd ‘𝑢) = 𝑥) |
59 | 58 | breq1d 5084 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → ((2nd ‘𝑢)𝑆(2nd ‘𝑣) ↔ 𝑥𝑆(2nd ‘𝑣))) |
60 | 57, 59 | anbi12d 631 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → (((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd
‘𝑢)𝑆(2nd ‘𝑣)) ↔ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣)))) |
61 | 56, 60 | orbi12d 916 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → (((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd
‘𝑢)𝑆(2nd ‘𝑣))) ↔ ((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣))))) |
62 | 54, 61 | anbi12d 631 |
. . . . . . . . . . . . 13
⊢ (𝑢 = 〈(𝐹‘𝑥), 𝑥〉 → (((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd
‘𝑢)𝑆(2nd ‘𝑣)))) ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣)))))) |
63 | | eleq1 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (𝑣 ∈ (𝐵 × 𝐴) ↔ 〈(𝐹‘𝑦), 𝑦〉 ∈ (𝐵 × 𝐴))) |
64 | | opelxp 5625 |
. . . . . . . . . . . . . . . 16
⊢
(〈(𝐹‘𝑦), 𝑦〉 ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) |
65 | 63, 64 | bitrdi 287 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (𝑣 ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))) |
66 | 65 | anbi2d 629 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)))) |
67 | | fvex 6787 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹‘𝑦) ∈ V |
68 | | vex 3436 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑦 ∈ V |
69 | 67, 68 | op1std 7841 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (1st ‘𝑣) = (𝐹‘𝑦)) |
70 | 69 | breq2d 5086 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → ((𝐹‘𝑥)𝑅(1st ‘𝑣) ↔ (𝐹‘𝑥)𝑅(𝐹‘𝑦))) |
71 | 69 | eqeq2d 2749 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → ((𝐹‘𝑥) = (1st ‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) |
72 | 67, 68 | op2ndd 7842 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (2nd ‘𝑣) = 𝑦) |
73 | 72 | breq2d 5086 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (𝑥𝑆(2nd ‘𝑣) ↔ 𝑥𝑆𝑦)) |
74 | 71, 73 | anbi12d 631 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣)) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))) |
75 | 70, 74 | orbi12d 916 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣))) ↔ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))) |
76 | 66, 75 | anbi12d 631 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 〈(𝐹‘𝑦), 𝑦〉 → (((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣)))) ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))))) |
77 | 19, 24, 62, 76, 11 | brab 5456 |
. . . . . . . . . . . 12
⊢
(〈(𝐹‘𝑥), 𝑥〉𝑄〈(𝐹‘𝑦), 𝑦〉 ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))) |
78 | | ffvelrn 6959 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
79 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) |
80 | 78, 79 | jca 512 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) |
81 | | ffvelrn 6959 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵) |
82 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) |
83 | 81, 82 | jca 512 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) |
84 | 80, 83 | anim12dan 619 |
. . . . . . . . . . . . 13
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))) |
85 | 84 | biantrurd 533 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))))) |
86 | 77, 85 | bitr4id 290 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (〈(𝐹‘𝑥), 𝑥〉𝑄〈(𝐹‘𝑦), 𝑦〉 ↔ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))) |
87 | 48, 50, 86 | 3bitrrd 306 |
. . . . . . . . . 10
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)) ↔ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))) |
88 | 87 | pm5.32da 579 |
. . . . . . . . 9
⊢ (𝐹:𝐴⟶𝐵 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦)))) |
89 | 88 | opabbidv 5140 |
. . . . . . . 8
⊢ (𝐹:𝐴⟶𝐵 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))}) |
90 | 40, 89 | eqtrid 2790 |
. . . . . . 7
⊢ (𝐹:𝐴⟶𝐵 → 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))}) |
91 | | isoeq3 7190 |
. . . . . . 7
⊢ (𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} → (◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) ↔ ◡𝐺 Isom 𝑄, {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} (ran 𝐺, 𝐴))) |
92 | 90, 91 | syl 17 |
. . . . . 6
⊢ (𝐹:𝐴⟶𝐵 → (◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) ↔ ◡𝐺 Isom 𝑄, {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} (ran 𝐺, 𝐴))) |
93 | 39, 92 | syl5ibr 245 |
. . . . 5
⊢ (𝐹:𝐴⟶𝐵 → (◡𝐺:ran 𝐺–1-1-onto→𝐴 → ◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴))) |
94 | 1, 37, 93 | sylc 65 |
. . . 4
⊢ (𝜑 → ◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴)) |
95 | | isocnv 7201 |
. . . 4
⊢ (◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) → ◡◡𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺)) |
96 | 94, 95 | syl 17 |
. . 3
⊢ (𝜑 → ◡◡𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺)) |
97 | | imacnvcnv 6109 |
. . . . 5
⊢ (◡◡𝐺 “ 𝑤) = (𝐺 “ 𝑤) |
98 | | fnwe.5 |
. . . . . . 7
⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V) |
99 | | vex 3436 |
. . . . . . 7
⊢ 𝑤 ∈ V |
100 | | xpexg 7600 |
. . . . . . 7
⊢ (((𝐹 “ 𝑤) ∈ V ∧ 𝑤 ∈ V) → ((𝐹 “ 𝑤) × 𝑤) ∈ V) |
101 | 98, 99, 100 | sylancl 586 |
. . . . . 6
⊢ (𝜑 → ((𝐹 “ 𝑤) × 𝑤) ∈ V) |
102 | | imadmres 6137 |
. . . . . . 7
⊢ (𝐺 “ dom (𝐺 ↾ 𝑤)) = (𝐺 “ 𝑤) |
103 | | dmres 5913 |
. . . . . . . . . . 11
⊢ dom
(𝐺 ↾ 𝑤) = (𝑤 ∩ dom 𝐺) |
104 | 103 | elin2 4131 |
. . . . . . . . . 10
⊢ (𝑥 ∈ dom (𝐺 ↾ 𝑤) ↔ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) |
105 | | simprr 770 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐺) |
106 | | f1dm 6674 |
. . . . . . . . . . . . . . 15
⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) → dom 𝐺 = 𝐴) |
107 | 1, 34, 106 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → dom 𝐺 = 𝐴) |
108 | 107 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → dom 𝐺 = 𝐴) |
109 | 105, 108 | eleqtrd 2841 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ 𝐴) |
110 | 109, 20 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐺‘𝑥) = 〈(𝐹‘𝑥), 𝑥〉) |
111 | 1 | ffnd 6601 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹 Fn 𝐴) |
112 | 111 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝐹 Fn 𝐴) |
113 | | dmres 5913 |
. . . . . . . . . . . . . . 15
⊢ dom
(𝐹 ↾ 𝑤) = (𝑤 ∩ dom 𝐹) |
114 | | inss2 4163 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∩ dom 𝐹) ⊆ dom 𝐹 |
115 | 112 | fndmd 6538 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → dom 𝐹 = 𝐴) |
116 | 114, 115 | sseqtrid 3973 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝑤 ∩ dom 𝐹) ⊆ 𝐴) |
117 | 113, 116 | eqsstrid 3969 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → dom (𝐹 ↾ 𝑤) ⊆ 𝐴) |
118 | | simprl 768 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ 𝑤) |
119 | 109, 115 | eleqtrrd 2842 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐹) |
120 | 113 | elin2 4131 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ dom (𝐹 ↾ 𝑤) ↔ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐹)) |
121 | 118, 119,
120 | sylanbrc 583 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom (𝐹 ↾ 𝑤)) |
122 | | fnfvima 7109 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 Fn 𝐴 ∧ dom (𝐹 ↾ 𝑤) ⊆ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝑤)) → (𝐹‘𝑥) ∈ (𝐹 “ dom (𝐹 ↾ 𝑤))) |
123 | 112, 117,
121, 122 | syl3anc 1370 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐹‘𝑥) ∈ (𝐹 “ dom (𝐹 ↾ 𝑤))) |
124 | | imadmres 6137 |
. . . . . . . . . . . . 13
⊢ (𝐹 “ dom (𝐹 ↾ 𝑤)) = (𝐹 “ 𝑤) |
125 | 123, 124 | eleqtrdi 2849 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑤)) |
126 | 125, 118 | opelxpd 5627 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 〈(𝐹‘𝑥), 𝑥〉 ∈ ((𝐹 “ 𝑤) × 𝑤)) |
127 | 110, 126 | eqeltrd 2839 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤)) |
128 | 104, 127 | sylan2b 594 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐺 ↾ 𝑤)) → (𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤)) |
129 | 128 | ralrimiva 3103 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ dom (𝐺 ↾ 𝑤)(𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤)) |
130 | | f1fun 6672 |
. . . . . . . . . 10
⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) → Fun 𝐺) |
131 | 1, 34, 130 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → Fun 𝐺) |
132 | | resss 5916 |
. . . . . . . . . 10
⊢ (𝐺 ↾ 𝑤) ⊆ 𝐺 |
133 | | dmss 5811 |
. . . . . . . . . 10
⊢ ((𝐺 ↾ 𝑤) ⊆ 𝐺 → dom (𝐺 ↾ 𝑤) ⊆ dom 𝐺) |
134 | 132, 133 | ax-mp 5 |
. . . . . . . . 9
⊢ dom
(𝐺 ↾ 𝑤) ⊆ dom 𝐺 |
135 | | funimass4 6834 |
. . . . . . . . 9
⊢ ((Fun
𝐺 ∧ dom (𝐺 ↾ 𝑤) ⊆ dom 𝐺) → ((𝐺 “ dom (𝐺 ↾ 𝑤)) ⊆ ((𝐹 “ 𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺 ↾ 𝑤)(𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤))) |
136 | 131, 134,
135 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 “ dom (𝐺 ↾ 𝑤)) ⊆ ((𝐹 “ 𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺 ↾ 𝑤)(𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤))) |
137 | 129, 136 | mpbird 256 |
. . . . . . 7
⊢ (𝜑 → (𝐺 “ dom (𝐺 ↾ 𝑤)) ⊆ ((𝐹 “ 𝑤) × 𝑤)) |
138 | 102, 137 | eqsstrrid 3970 |
. . . . . 6
⊢ (𝜑 → (𝐺 “ 𝑤) ⊆ ((𝐹 “ 𝑤) × 𝑤)) |
139 | 101, 138 | ssexd 5248 |
. . . . 5
⊢ (𝜑 → (𝐺 “ 𝑤) ∈ V) |
140 | 97, 139 | eqeltrid 2843 |
. . . 4
⊢ (𝜑 → (◡◡𝐺 “ 𝑤) ∈ V) |
141 | 140 | alrimiv 1930 |
. . 3
⊢ (𝜑 → ∀𝑤(◡◡𝐺 “ 𝑤) ∈ V) |
142 | | isowe2 7221 |
. . 3
⊢ ((◡◡𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺) ∧ ∀𝑤(◡◡𝐺 “ 𝑤) ∈ V) → (𝑄 We ran 𝐺 → 𝑇 We 𝐴)) |
143 | 96, 141, 142 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝑄 We ran 𝐺 → 𝑇 We 𝐴)) |
144 | 15, 143 | mpd 15 |
1
⊢ (𝜑 → 𝑇 We 𝐴) |