Step | Hyp | Ref
| Expression |
1 | | opeq1 4835 |
. . . 4
⊢ (𝑎 = 𝑋 → ⟨𝑎, 𝑏⟩ = ⟨𝑋, 𝑏⟩) |
2 | | predeq3 6262 |
. . . 4
⊢
(⟨𝑎, 𝑏⟩ = ⟨𝑋, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑏⟩)) |
3 | 1, 2 | syl 17 |
. . 3
⊢ (𝑎 = 𝑋 → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑏⟩)) |
4 | | predeq3 6262 |
. . . . . 6
⊢ (𝑎 = 𝑋 → Pred(𝑅, 𝐴, 𝑎) = Pred(𝑅, 𝐴, 𝑋)) |
5 | | sneq 4601 |
. . . . . 6
⊢ (𝑎 = 𝑋 → {𝑎} = {𝑋}) |
6 | 4, 5 | uneq12d 4129 |
. . . . 5
⊢ (𝑎 = 𝑋 → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) = (Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋})) |
7 | 6 | xpeq1d 5667 |
. . . 4
⊢ (𝑎 = 𝑋 → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) = ((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}))) |
8 | 1 | sneqd 4603 |
. . . 4
⊢ (𝑎 = 𝑋 → {⟨𝑎, 𝑏⟩} = {⟨𝑋, 𝑏⟩}) |
9 | 7, 8 | difeq12d 4088 |
. . 3
⊢ (𝑎 = 𝑋 → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑋, 𝑏⟩})) |
10 | 3, 9 | eqeq12d 2753 |
. 2
⊢ (𝑎 = 𝑋 → (Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ↔ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑋, 𝑏⟩}))) |
11 | | opeq2 4836 |
. . . 4
⊢ (𝑏 = 𝑌 → ⟨𝑋, 𝑏⟩ = ⟨𝑋, 𝑌⟩) |
12 | | predeq3 6262 |
. . . 4
⊢
(⟨𝑋, 𝑏⟩ = ⟨𝑋, 𝑌⟩ → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑏⟩) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩)) |
13 | 11, 12 | syl 17 |
. . 3
⊢ (𝑏 = 𝑌 → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑏⟩) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩)) |
14 | | predeq3 6262 |
. . . . . 6
⊢ (𝑏 = 𝑌 → Pred(𝑆, 𝐵, 𝑏) = Pred(𝑆, 𝐵, 𝑌)) |
15 | | sneq 4601 |
. . . . . 6
⊢ (𝑏 = 𝑌 → {𝑏} = {𝑌}) |
16 | 14, 15 | uneq12d 4129 |
. . . . 5
⊢ (𝑏 = 𝑌 → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) = (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) |
17 | 16 | xpeq2d 5668 |
. . . 4
⊢ (𝑏 = 𝑌 → ((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) = ((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌}))) |
18 | 11 | sneqd 4603 |
. . . 4
⊢ (𝑏 = 𝑌 → {⟨𝑋, 𝑏⟩} = {⟨𝑋, 𝑌⟩}) |
19 | 17, 18 | difeq12d 4088 |
. . 3
⊢ (𝑏 = 𝑌 → (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑋, 𝑏⟩}) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) ∖ {⟨𝑋, 𝑌⟩})) |
20 | 13, 19 | eqeq12d 2753 |
. 2
⊢ (𝑏 = 𝑌 → (Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑋, 𝑏⟩}) ↔ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) ∖ {⟨𝑋, 𝑌⟩}))) |
21 | | predel 6279 |
. . . . 5
⊢ (𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) → 𝑒 ∈ (𝐴 × 𝐵)) |
22 | 21 | a1i 11 |
. . . 4
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) → 𝑒 ∈ (𝐴 × 𝐵))) |
23 | | eldifi 4091 |
. . . . 5
⊢ (𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) → 𝑒 ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}))) |
24 | | predss 6266 |
. . . . . . . . 9
⊢
Pred(𝑅, 𝐴, 𝑎) ⊆ 𝐴 |
25 | 24 | a1i 11 |
. . . . . . . 8
⊢ (𝑎 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝐴) |
26 | | snssi 4773 |
. . . . . . . 8
⊢ (𝑎 ∈ 𝐴 → {𝑎} ⊆ 𝐴) |
27 | 25, 26 | unssd 4151 |
. . . . . . 7
⊢ (𝑎 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ⊆ 𝐴) |
28 | | predss 6266 |
. . . . . . . . 9
⊢
Pred(𝑆, 𝐵, 𝑏) ⊆ 𝐵 |
29 | 28 | a1i 11 |
. . . . . . . 8
⊢ (𝑏 ∈ 𝐵 → Pred(𝑆, 𝐵, 𝑏) ⊆ 𝐵) |
30 | | snssi 4773 |
. . . . . . . 8
⊢ (𝑏 ∈ 𝐵 → {𝑏} ⊆ 𝐵) |
31 | 29, 30 | unssd 4151 |
. . . . . . 7
⊢ (𝑏 ∈ 𝐵 → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ⊆ 𝐵) |
32 | | xpss12 5653 |
. . . . . . 7
⊢
(((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ⊆ 𝐴 ∧ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ⊆ 𝐵) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ⊆ (𝐴 × 𝐵)) |
33 | 27, 31, 32 | syl2an 597 |
. . . . . 6
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ⊆ (𝐴 × 𝐵)) |
34 | 33 | sseld 3948 |
. . . . 5
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑒 ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) → 𝑒 ∈ (𝐴 × 𝐵))) |
35 | 23, 34 | syl5 34 |
. . . 4
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) → 𝑒 ∈ (𝐴 × 𝐵))) |
36 | | elxp2 5662 |
. . . . 5
⊢ (𝑒 ∈ (𝐴 × 𝐵) ↔ ∃𝑐 ∈ 𝐴 ∃𝑑 ∈ 𝐵 𝑒 = ⟨𝑐, 𝑑⟩) |
37 | | opex 5426 |
. . . . . . . . 9
⊢
⟨𝑎, 𝑏⟩ ∈ V |
38 | | opex 5426 |
. . . . . . . . . 10
⊢
⟨𝑐, 𝑑⟩ ∈ V |
39 | 38 | elpred 6275 |
. . . . . . . . 9
⊢
(⟨𝑎, 𝑏⟩ ∈ V →
(⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑐, 𝑑⟩𝑇⟨𝑎, 𝑏⟩))) |
40 | 37, 39 | ax-mp 5 |
. . . . . . . 8
⊢
(⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑐, 𝑑⟩𝑇⟨𝑎, 𝑏⟩)) |
41 | | opelxpi 5675 |
. . . . . . . . . . 11
⊢ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) → ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)) |
42 | 41 | adantl 483 |
. . . . . . . . . 10
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)) |
43 | 42 | biantrurd 534 |
. . . . . . . . 9
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (⟨𝑐, 𝑑⟩𝑇⟨𝑎, 𝑏⟩ ↔ (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑐, 𝑑⟩𝑇⟨𝑎, 𝑏⟩))) |
44 | | xpord2.1 |
. . . . . . . . . . 11
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd
‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} |
45 | 44 | xpord2lem 8079 |
. . . . . . . . . 10
⊢
(⟨𝑐, 𝑑⟩𝑇⟨𝑎, 𝑏⟩ ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))) |
46 | | eldif 3925 |
. . . . . . . . . . . . 13
⊢
(⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ↔ (⟨𝑐, 𝑑⟩ ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ ¬ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩})) |
47 | | opelxp 5674 |
. . . . . . . . . . . . . . 15
⊢
(⟨𝑐, 𝑑⟩ ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ↔ (𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}))) |
48 | | elun 4113 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ↔ (𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) ∨ 𝑐 ∈ {𝑎})) |
49 | | vex 3452 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑐 ∈ V |
50 | 49 | elpred 6275 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ∈ V → (𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) ↔ (𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎))) |
51 | 50 | elv 3454 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) ↔ (𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎)) |
52 | | velsn 4607 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 ∈ {𝑎} ↔ 𝑐 = 𝑎) |
53 | 51, 52 | orbi12i 914 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑐 ∈ Pred(𝑅, 𝐴, 𝑎) ∨ 𝑐 ∈ {𝑎}) ↔ ((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎)) |
54 | 48, 53 | bitri 275 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ↔ ((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎)) |
55 | | elun 4113 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ↔ (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) ∨ 𝑑 ∈ {𝑏})) |
56 | | vex 3452 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑑 ∈ V |
57 | 56 | elpred 6275 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 ∈ V → (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) ↔ (𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏))) |
58 | 57 | elv 3454 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) ↔ (𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏)) |
59 | | velsn 4607 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑑 ∈ {𝑏} ↔ 𝑑 = 𝑏) |
60 | 58, 59 | orbi12i 914 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑑 ∈ Pred(𝑆, 𝐵, 𝑏) ∨ 𝑑 ∈ {𝑏}) ↔ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏)) |
61 | 55, 60 | bitri 275 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ↔ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏)) |
62 | 54, 61 | anbi12i 628 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐 ∈ (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∧ 𝑑 ∈ (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ↔ (((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏))) |
63 | 47, 62 | bitri 275 |
. . . . . . . . . . . . . 14
⊢
(⟨𝑐, 𝑑⟩ ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ↔ (((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏))) |
64 | 38 | elsn 4606 |
. . . . . . . . . . . . . . . 16
⊢
(⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩} ↔ ⟨𝑐, 𝑑⟩ = ⟨𝑎, 𝑏⟩) |
65 | 64 | notbii 320 |
. . . . . . . . . . . . . . 15
⊢ (¬
⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩} ↔ ¬ ⟨𝑐, 𝑑⟩ = ⟨𝑎, 𝑏⟩) |
66 | | df-ne 2945 |
. . . . . . . . . . . . . . 15
⊢
(⟨𝑐, 𝑑⟩ ≠ ⟨𝑎, 𝑏⟩ ↔ ¬ ⟨𝑐, 𝑑⟩ = ⟨𝑎, 𝑏⟩) |
67 | 49, 56 | opthne 5444 |
. . . . . . . . . . . . . . 15
⊢
(⟨𝑐, 𝑑⟩ ≠ ⟨𝑎, 𝑏⟩ ↔ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) |
68 | 65, 66, 67 | 3bitr2i 299 |
. . . . . . . . . . . . . 14
⊢ (¬
⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩} ↔ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) |
69 | 63, 68 | anbi12i 628 |
. . . . . . . . . . . . 13
⊢
((⟨𝑐, 𝑑⟩ ∈ ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∧ ¬ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑎, 𝑏⟩}) ↔ ((((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏)) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))) |
70 | 46, 69 | bitri 275 |
. . . . . . . . . . . 12
⊢
(⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ↔ ((((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏)) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))) |
71 | | simprl 770 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → 𝑐 ∈ 𝐴) |
72 | 71 | biantrurd 534 |
. . . . . . . . . . . . . . 15
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (𝑐𝑅𝑎 ↔ (𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎))) |
73 | 72 | orbi1d 916 |
. . . . . . . . . . . . . 14
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ↔ ((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎))) |
74 | | simprr 772 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → 𝑑 ∈ 𝐵) |
75 | 74 | biantrurd 534 |
. . . . . . . . . . . . . . 15
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (𝑑𝑆𝑏 ↔ (𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏))) |
76 | 75 | orbi1d 916 |
. . . . . . . . . . . . . 14
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → ((𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ↔ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏))) |
77 | 73, 76 | 3anbi12d 1438 |
. . . . . . . . . . . . 13
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) ↔ (((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))) |
78 | | df-3an 1090 |
. . . . . . . . . . . . 13
⊢ ((((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) ↔ ((((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏)) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))) |
79 | 77, 78 | bitr2di 288 |
. . . . . . . . . . . 12
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (((((𝑐 ∈ 𝐴 ∧ 𝑐𝑅𝑎) ∨ 𝑐 = 𝑎) ∧ ((𝑑 ∈ 𝐵 ∧ 𝑑𝑆𝑏) ∨ 𝑑 = 𝑏)) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) ↔ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))) |
80 | 70, 79 | bitrid 283 |
. . . . . . . . . . 11
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ↔ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))) |
81 | | pm3.22 461 |
. . . . . . . . . . . . 13
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) |
82 | 81 | biantrurd 534 |
. . . . . . . . . . . 12
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) ↔ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))))) |
83 | | df-3an 1090 |
. . . . . . . . . . . 12
⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))) ↔ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)))) |
84 | 82, 83 | bitr4di 289 |
. . . . . . . . . . 11
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏)) ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))))) |
85 | 80, 84 | bitr2d 280 |
. . . . . . . . . 10
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ ((𝑐𝑅𝑎 ∨ 𝑐 = 𝑎) ∧ (𝑑𝑆𝑏 ∨ 𝑑 = 𝑏) ∧ (𝑐 ≠ 𝑎 ∨ 𝑑 ≠ 𝑏))) ↔ ⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))) |
86 | 45, 85 | bitrid 283 |
. . . . . . . . 9
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (⟨𝑐, 𝑑⟩𝑇⟨𝑎, 𝑏⟩ ↔ ⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))) |
87 | 43, 86 | bitr3d 281 |
. . . . . . . 8
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → ((⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑐, 𝑑⟩𝑇⟨𝑎, 𝑏⟩) ↔ ⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))) |
88 | 40, 87 | bitrid 283 |
. . . . . . 7
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ ⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))) |
89 | | eleq1 2826 |
. . . . . . . 8
⊢ (𝑒 = ⟨𝑐, 𝑑⟩ → (𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ ⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))) |
90 | | eleq1 2826 |
. . . . . . . 8
⊢ (𝑒 = ⟨𝑐, 𝑑⟩ → (𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ↔ ⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))) |
91 | 89, 90 | bibi12d 346 |
. . . . . . 7
⊢ (𝑒 = ⟨𝑐, 𝑑⟩ → ((𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ 𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩})) ↔ (⟨𝑐, 𝑑⟩ ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ ⟨𝑐, 𝑑⟩ ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩})))) |
92 | 88, 91 | syl5ibrcom 247 |
. . . . . 6
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) → (𝑒 = ⟨𝑐, 𝑑⟩ → (𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ 𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩})))) |
93 | 92 | rexlimdvva 3206 |
. . . . 5
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (∃𝑐 ∈ 𝐴 ∃𝑑 ∈ 𝐵 𝑒 = ⟨𝑐, 𝑑⟩ → (𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ 𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩})))) |
94 | 36, 93 | biimtrid 241 |
. . . 4
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑒 ∈ (𝐴 × 𝐵) → (𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ 𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩})))) |
95 | 22, 35, 94 | pm5.21ndd 381 |
. . 3
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑒 ∈ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ↔ 𝑒 ∈ (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))) |
96 | 95 | eqrdv 2735 |
. 2
⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩})) |
97 | 10, 20, 96 | vtocl2ga 3538 |
1
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) ∖ {⟨𝑋, 𝑌⟩})) |