Step | Hyp | Ref
| Expression |
1 | | ssun2 4107 |
. . . 4
⊢ {(𝑋𝐹(𝑊‘𝑋))} ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
2 | | fpwwe2.1 |
. . . . . . . . . . . . . 14
⊢ 𝑊 = {〈𝑥, 𝑟〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} |
3 | | fpwwe2.2 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
4 | 3 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝐴 ∈ 𝑉) |
5 | | fpwwe2.3 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) |
6 | 5 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) |
7 | | fpwwe2.4 |
. . . . . . . . . . . . . 14
⊢ 𝑋 = ∪
dom 𝑊 |
8 | 2, 4, 6, 7 | fpwwe2lem11 10397 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑋 ∈ dom 𝑊) |
9 | 2, 4, 6, 7 | fpwwe2lem10 10396 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋)) |
10 | | ffun 6603 |
. . . . . . . . . . . . . 14
⊢ (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) → Fun 𝑊) |
11 | | funfvbrb 6928 |
. . . . . . . . . . . . . 14
⊢ (Fun
𝑊 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋))) |
12 | 9, 10, 11 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋))) |
13 | 8, 12 | mpbid 231 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑋𝑊(𝑊‘𝑋)) |
14 | 2, 4 | fpwwe2lem2 10388 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋𝑊(𝑊‘𝑋) ↔ ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))) |
15 | 13, 14 | mpbid 231 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))) |
16 | 15 | simpld 495 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋))) |
17 | 16 | simpld 495 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑋 ⊆ 𝐴) |
18 | 16 | simprd 496 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
19 | 15 | simprd 496 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)) |
20 | 19 | simpld 495 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) We 𝑋) |
21 | 17, 18, 20 | 3jca 1127 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋) ∧ (𝑊‘𝑋) We 𝑋)) |
22 | 2, 3, 5 | fpwwe2lem4 10390 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋) ∧ (𝑊‘𝑋) We 𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝐴) |
23 | 21, 22 | syldan 591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝐴) |
24 | 23 | snssd 4742 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → {(𝑋𝐹(𝑊‘𝑋))} ⊆ 𝐴) |
25 | 17, 24 | unssd 4120 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴) |
26 | | ssun1 4106 |
. . . . . . . . . . 11
⊢ 𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
27 | | xpss12 5604 |
. . . . . . . . . . 11
⊢ ((𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑋 × 𝑋) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
28 | 26, 26, 27 | mp2an 689 |
. . . . . . . . . 10
⊢ (𝑋 × 𝑋) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
29 | 18, 28 | sstrdi 3933 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
30 | | xpss12 5604 |
. . . . . . . . . . 11
⊢ ((𝑋 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ {(𝑋𝐹(𝑊‘𝑋))} ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
31 | 26, 1, 30 | mp2an 689 |
. . . . . . . . . 10
⊢ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
32 | 31 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
33 | 29, 32 | unssd 4120 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) |
34 | 25, 33 | jca 512 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴 ∧ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})))) |
35 | | ssdif0 4297 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))} ↔ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) = ∅) |
36 | | simpllr 773 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
37 | 18 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
38 | 37 | ssbrd 5117 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)(𝑋𝐹(𝑊‘𝑋)))) |
39 | | brxp 5636 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)(𝑋𝐹(𝑊‘𝑋)) ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
40 | 39 | simplbi 498 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
41 | 38, 40 | syl6 35 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
42 | 36, 41 | mtod 197 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋))) |
43 | | brxp 5636 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)) ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
44 | 43 | simplbi 498 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
45 | 36, 44 | nsyl 140 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))) |
46 | | ovex 7308 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑋𝐹(𝑊‘𝑋)) ∈ V |
47 | | breq2 5078 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑋𝐹(𝑊‘𝑋)) → ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))(𝑋𝐹(𝑊‘𝑋)))) |
48 | | brun 5125 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))(𝑋𝐹(𝑊‘𝑋)) ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
49 | 47, 48 | bitrdi 287 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑋𝐹(𝑊‘𝑋)) → ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))))) |
50 | 49 | notbid 318 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑋𝐹(𝑊‘𝑋)) → (¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))))) |
51 | 46, 50 | rexsn 4618 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑦 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
52 | | ioran 981 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋))) ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
53 | 51, 52 | bitri 274 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑦 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)(𝑋𝐹(𝑊‘𝑋)) ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})(𝑋𝐹(𝑊‘𝑋)))) |
54 | 42, 45, 53 | sylanbrc 583 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ∃𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
55 | | simpr 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) |
56 | | sssn 4759 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))} ↔ (𝑥 = ∅ ∨ 𝑥 = {(𝑋𝐹(𝑊‘𝑋))})) |
57 | 55, 56 | sylib 217 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥 = ∅ ∨ 𝑥 = {(𝑋𝐹(𝑊‘𝑋))})) |
58 | | simplrr 775 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 ≠ ∅) |
59 | 58 | neneqd 2948 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ 𝑥 = ∅) |
60 | 57, 59 | orcnd 876 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 = {(𝑋𝐹(𝑊‘𝑋))}) |
61 | 60 | raleqdv 3348 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
62 | | breq1 5077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
63 | 62 | notbid 318 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
64 | 46, 63 | ralsn 4617 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑧 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
65 | 61, 64 | bitrdi 287 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
66 | 60, 65 | rexeqbidv 3337 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → (∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ∃𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ (𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
67 | 54, 66 | mpbird 256 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ 𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
68 | 67 | ex 413 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → (𝑥 ⊆ {(𝑋𝐹(𝑊‘𝑋))} → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
69 | 35, 68 | syl5bir 242 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) = ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
70 | | vex 3436 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑥 ∈ V |
71 | | difexg 5251 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ V → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
72 | 70, 71 | mp1i 13 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
73 | | wefr 5579 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Fr 𝑋) |
74 | 20, 73 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) Fr 𝑋) |
75 | 74 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑊‘𝑋) Fr 𝑋) |
76 | | simplrl 774 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → 𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
77 | | uncom 4087 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} ∪ 𝑋) |
78 | 76, 77 | sseqtrdi 3971 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → 𝑥 ⊆ ({(𝑋𝐹(𝑊‘𝑋))} ∪ 𝑋)) |
79 | | ssundif 4418 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ⊆ ({(𝑋𝐹(𝑊‘𝑋))} ∪ 𝑋) ↔ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋) |
80 | 78, 79 | sylib 217 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋) |
81 | | simpr 485 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) |
82 | | fri 5549 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋 ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅)) → ∃𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦) |
83 | 72, 75, 80, 81, 82 | syl22anc 836 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → ∃𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦) |
84 | | brun 5125 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (𝑧(𝑊‘𝑋)𝑦 ∨ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
85 | | idd 24 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑧(𝑊‘𝑋)𝑦 → 𝑧(𝑊‘𝑋)𝑦)) |
86 | | brxp 5636 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ↔ (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
87 | 86 | simprbi 497 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
88 | | eldifn 4062 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
89 | 88 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
90 | 89 | pm2.21d 121 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} → 𝑧(𝑊‘𝑋)𝑦)) |
91 | 87, 90 | syl5 34 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑧(𝑊‘𝑋)𝑦)) |
92 | 85, 91 | jaod 856 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ((𝑧(𝑊‘𝑋)𝑦 ∨ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) → 𝑧(𝑊‘𝑋)𝑦)) |
93 | 84, 92 | syl5bi 241 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 → 𝑧(𝑊‘𝑋)𝑦)) |
94 | 93 | con3d 152 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (¬ 𝑧(𝑊‘𝑋)𝑦 → ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
95 | 94 | ralimdv 3109 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → ∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
96 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
97 | 96 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
98 | 18 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
99 | 98 | ssbrd 5117 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 → (𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)𝑦)) |
100 | | brxp 5636 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) |
101 | 100 | simplbi 498 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × 𝑋)𝑦 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
102 | 99, 101 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
103 | 97, 102 | mtod 197 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦) |
104 | | brxp 5636 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
105 | 104 | simprbi 497 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) |
106 | 89, 105 | nsyl 140 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) |
107 | | brun 5125 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑋𝐹(𝑊‘𝑋))((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
108 | 62, 107 | bitrdi 287 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦))) |
109 | 108 | notbid 318 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 = (𝑋𝐹(𝑊‘𝑋)) → (¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦))) |
110 | 46, 109 | ralsn 4617 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑧 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ ¬ ((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
111 | | ioran 981 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
((𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∨ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
112 | 110, 111 | bitri 274 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑧 ∈
{(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ (¬ (𝑋𝐹(𝑊‘𝑋))(𝑊‘𝑋)𝑦 ∧ ¬ (𝑋𝐹(𝑊‘𝑋))(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
113 | 103, 106,
112 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
114 | 95, 113 | jctird 527 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∧ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦))) |
115 | | ssun1 4106 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑥 ⊆ (𝑥 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
116 | | undif1 4409 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) = (𝑥 ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
117 | 115, 116 | sseqtrri 3958 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑥 ⊆ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) |
118 | | ralun 4126 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑧 ∈
(𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∧ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) → ∀𝑧 ∈ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
119 | | ssralv 3987 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ⊆ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) → (∀𝑧 ∈ ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∪ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 → ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
120 | 117, 118,
119 | mpsyl 68 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑧 ∈
(𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∧ ∀𝑧 ∈ {(𝑋𝐹(𝑊‘𝑋))} ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) → ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
121 | 114, 120 | syl6 35 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
122 | | eldifi 4061 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑦 ∈ 𝑥) |
123 | 122 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → 𝑦 ∈ 𝑥) |
124 | 121, 123 | jctild 526 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ ¬
(𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) ∧ 𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})) → (∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦))) |
125 | 124 | expimpd 454 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → ((𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦) → (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦))) |
126 | 125 | reximdv2 3199 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → (∃𝑦 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))})∀𝑧 ∈ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ¬ 𝑧(𝑊‘𝑋)𝑦 → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
127 | 83, 126 | mpd 15 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) ∧ (𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
128 | 127 | ex 413 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → ((𝑥 ∖ {(𝑋𝐹(𝑊‘𝑋))}) ≠ ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
129 | 69, 128 | pm2.61dne 3031 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅)) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
130 | 129 | ex 413 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
131 | 130 | alrimiv 1930 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ∀𝑥((𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
132 | | df-fr 5544 |
. . . . . . . . . 10
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) Fr (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ ∀𝑥((𝑥 ⊆ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
133 | 131, 132 | sylibr 233 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) Fr (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
134 | | elun 4083 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
135 | | elun 4083 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
136 | 134, 135 | anbi12i 627 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) ↔ ((𝑥 ∈ 𝑋 ∨ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}))) |
137 | | weso 5580 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Or 𝑋) |
138 | 20, 137 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑊‘𝑋) Or 𝑋) |
139 | | solin 5528 |
. . . . . . . . . . . . . . 15
⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(𝑊‘𝑋)𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑊‘𝑋)𝑥)) |
140 | 138, 139 | sylan 580 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(𝑊‘𝑋)𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑊‘𝑋)𝑥)) |
141 | | ssun1 4106 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊‘𝑋) ⊆ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
142 | 141 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑊‘𝑋) ⊆ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))) |
143 | 142 | ssbrd 5117 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(𝑊‘𝑋)𝑦 → 𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
144 | | idd 24 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 = 𝑦 → 𝑥 = 𝑦)) |
145 | 142 | ssbrd 5117 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦(𝑊‘𝑋)𝑥 → 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
146 | 143, 144,
145 | 3orim123d 1443 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥(𝑊‘𝑋)𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑊‘𝑋)𝑥) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
147 | 140, 146 | mpd 15 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
148 | 147 | ex 413 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
149 | | simpr 485 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) |
150 | 149 | ancomd 462 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
151 | | brxp 5636 |
. . . . . . . . . . . . . . 15
⊢ (𝑦(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
152 | 150, 151 | sylibr 233 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → 𝑦(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑥) |
153 | | ssun2 4107 |
. . . . . . . . . . . . . . 15
⊢ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
154 | 153 | ssbri 5119 |
. . . . . . . . . . . . . 14
⊢ (𝑦(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑥 → 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥) |
155 | | 3mix3 1331 |
. . . . . . . . . . . . . 14
⊢ (𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥 → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
156 | 152, 154,
155 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋)) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
157 | 156 | ex 413 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ 𝑋) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
158 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
159 | | brxp 5636 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) |
160 | 158, 159 | sylibr 233 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → 𝑥(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) |
161 | 153 | ssbri 5119 |
. . . . . . . . . . . . . 14
⊢ (𝑥(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
162 | | 3mix1 1329 |
. . . . . . . . . . . . . 14
⊢ (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
163 | 160, 161,
162 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
164 | 163 | ex 413 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
165 | | elsni 4578 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} → 𝑥 = (𝑋𝐹(𝑊‘𝑋))) |
166 | | elsni 4578 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))} → 𝑦 = (𝑋𝐹(𝑊‘𝑋))) |
167 | | eqtr3 2764 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 = (𝑋𝐹(𝑊‘𝑋)) ∧ 𝑦 = (𝑋𝐹(𝑊‘𝑋))) → 𝑥 = 𝑦) |
168 | 165, 166,
167 | syl2an 596 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑥 = 𝑦) |
169 | 168 | 3mix2d 1336 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
170 | 169 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))} ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
171 | 148, 157,
164, 170 | ccased 1036 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (((𝑥 ∈ 𝑋 ∨ 𝑥 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
172 | 136, 171 | syl5bi 241 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → (𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
173 | 172 | ralrimivv 3122 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ∀𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})(𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥)) |
174 | | dfwe2 7624 |
. . . . . . . . 9
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ↔ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) Fr (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑥 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})(𝑥((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑥))) |
175 | 133, 173,
174 | sylanbrc 583 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) |
176 | 2 | fpwwe2cbv 10386 |
. . . . . . . . . . . . 13
⊢ 𝑊 = {〈𝑎, 𝑠〉 ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑏](𝑏𝐹(𝑠 ∩ (𝑏 × 𝑏))) = 𝑧))} |
177 | 176, 4, 13 | fpwwe2lem3 10389 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = 𝑦) |
178 | | cnvimass 5989 |
. . . . . . . . . . . . . . 15
⊢ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ⊆ dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
179 | | fvex 6787 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊‘𝑋) ∈ V |
180 | | snex 5354 |
. . . . . . . . . . . . . . . . . 18
⊢ {(𝑋𝐹(𝑊‘𝑋))} ∈ V |
181 | | xpexg 7600 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑋 ∈ dom 𝑊 ∧ {(𝑋𝐹(𝑊‘𝑋))} ∈ V) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
182 | 8, 180, 181 | sylancl 586 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) |
183 | | unexg 7599 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑊‘𝑋) ∈ V ∧ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∈ V) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) |
184 | 179, 182,
183 | sylancr 587 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) |
185 | 184 | dmexd 7752 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) |
186 | | ssexg 5247 |
. . . . . . . . . . . . . . 15
⊢ (((◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ⊆ dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∧ dom ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∈ V) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
187 | 178, 185,
186 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
188 | 187 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
189 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) → 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) |
190 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋) |
191 | | simplr 766 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
192 | | nelne2 3042 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ 𝑋 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → 𝑦 ≠ (𝑋𝐹(𝑊‘𝑋))) |
193 | 190, 191,
192 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ≠ (𝑋𝐹(𝑊‘𝑋))) |
194 | 87, 166 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → 𝑦 = (𝑋𝐹(𝑊‘𝑋))) |
195 | 194 | necon3ai 2968 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ≠ (𝑋𝐹(𝑊‘𝑋)) → ¬ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦) |
196 | | biorf 934 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 → (𝑧(𝑊‘𝑋)𝑦 ↔ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦))) |
197 | 193, 195,
196 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑧(𝑊‘𝑋)𝑦 ↔ (𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦))) |
198 | | orcom 867 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦) ↔ (𝑧(𝑊‘𝑋)𝑦 ∨ 𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦)) |
199 | 198, 84 | bitr4i 277 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑧(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})𝑦 ∨ 𝑧(𝑊‘𝑋)𝑦) ↔ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
200 | 197, 199 | bitr2di 288 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦 ↔ 𝑧(𝑊‘𝑋)𝑦)) |
201 | | vex 3436 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑧 ∈ V |
202 | 201 | eliniseg 6002 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ V → (𝑧 ∈ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ↔ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦)) |
203 | 202 | elv 3438 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ↔ 𝑧((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))𝑦) |
204 | 201 | eliniseg 6002 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ V → (𝑧 ∈ (◡(𝑊‘𝑋) “ {𝑦}) ↔ 𝑧(𝑊‘𝑋)𝑦)) |
205 | 204 | elv 3438 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (◡(𝑊‘𝑋) “ {𝑦}) ↔ 𝑧(𝑊‘𝑋)𝑦) |
206 | 200, 203,
205 | 3bitr4g 314 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ↔ 𝑧 ∈ (◡(𝑊‘𝑋) “ {𝑦}))) |
207 | 206 | eqrdv 2736 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = (◡(𝑊‘𝑋) “ {𝑦})) |
208 | 189, 207 | sylan9eqr 2800 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → 𝑢 = (◡(𝑊‘𝑋) “ {𝑦})) |
209 | 208 | sqxpeqd 5621 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢 × 𝑢) = ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) |
210 | 209 | ineq2d 4146 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
211 | | indir 4209 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
212 | | inxp 5741 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦}))) |
213 | | incom 4135 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦})) = ((◡(𝑊‘𝑋) “ {𝑦}) ∩ {(𝑋𝐹(𝑊‘𝑋))}) |
214 | | cnvimass 5989 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (◡(𝑊‘𝑋) “ {𝑦}) ⊆ dom (𝑊‘𝑋) |
215 | 18 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
216 | | dmss 5811 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋)) |
217 | 215, 216 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋)) |
218 | | dmxpid 5839 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
219 | 217, 218 | sseqtrdi 3971 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → dom (𝑊‘𝑋) ⊆ 𝑋) |
220 | 214, 219 | sstrid 3932 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (◡(𝑊‘𝑋) “ {𝑦}) ⊆ 𝑋) |
221 | 220, 191 | ssneldd 3924 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ (◡(𝑊‘𝑋) “ {𝑦})) |
222 | | disjsn 4647 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((◡(𝑊‘𝑋) “ {𝑦}) ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅ ↔ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ (◡(𝑊‘𝑋) “ {𝑦})) |
223 | 221, 222 | sylibr 233 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((◡(𝑊‘𝑋) “ {𝑦}) ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅) |
224 | 213, 223 | eqtrid 2790 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦})) = ∅) |
225 | 224 | xpeq2d 5619 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ∅)) |
226 | | xp0 6061 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ∅) =
∅ |
227 | 225, 226 | eqtrdi 2794 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑦})) × ({(𝑋𝐹(𝑊‘𝑋))} ∩ (◡(𝑊‘𝑋) “ {𝑦}))) = ∅) |
228 | 212, 227 | eqtrid 2790 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ∅) |
229 | 228 | uneq2d 4097 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ∅)) |
230 | 211, 229 | eqtrid 2790 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ∅)) |
231 | | un0 4324 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) ∪ ∅) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) |
232 | 230, 231 | eqtrdi 2794 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
233 | 232 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
234 | 210, 233 | eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) |
235 | 208, 234 | oveq12d 7293 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦}))))) |
236 | 235 | eqeq1d 2740 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → ((𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = 𝑦)) |
237 | 188, 236 | sbcied 3761 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ([(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ((◡(𝑊‘𝑋) “ {𝑦})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑦}) × (◡(𝑊‘𝑋) “ {𝑦})))) = 𝑦)) |
238 | 177, 237 | mpbird 256 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
239 | 166 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → 𝑦 = (𝑋𝐹(𝑊‘𝑋))) |
240 | 239 | eqcomd 2744 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑋𝐹(𝑊‘𝑋)) = 𝑦) |
241 | 187 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) ∈ V) |
242 | | simplr 766 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
243 | 239 | eleq1d 2823 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑦 ∈ dom ◡(𝑊‘𝑋) ↔ (𝑋𝐹(𝑊‘𝑋)) ∈ dom ◡(𝑊‘𝑋))) |
244 | 18 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
245 | | rnss 5848 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) → ran (𝑊‘𝑋) ⊆ ran (𝑋 × 𝑋)) |
246 | 244, 245 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ran (𝑊‘𝑋) ⊆ ran (𝑋 × 𝑋)) |
247 | | df-rn 5600 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ran
(𝑊‘𝑋) = dom ◡(𝑊‘𝑋) |
248 | | rnxpid 6076 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ran
(𝑋 × 𝑋) = 𝑋 |
249 | 246, 247,
248 | 3sstr3g 3965 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → dom ◡(𝑊‘𝑋) ⊆ 𝑋) |
250 | 249 | sseld 3920 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋𝐹(𝑊‘𝑋)) ∈ dom ◡(𝑊‘𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
251 | 243, 250 | sylbid 239 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑦 ∈ dom ◡(𝑊‘𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
252 | 242, 251 | mtod 197 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ¬ 𝑦 ∈ dom ◡(𝑊‘𝑋)) |
253 | | ndmima 6011 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬
𝑦 ∈ dom ◡(𝑊‘𝑋) → (◡(𝑊‘𝑋) “ {𝑦}) = ∅) |
254 | 252, 253 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡(𝑊‘𝑋) “ {𝑦}) = ∅) |
255 | 239 | sneqd 4573 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → {𝑦} = {(𝑋𝐹(𝑊‘𝑋))}) |
256 | 255 | imaeq2d 5969 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦}) = (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {(𝑋𝐹(𝑊‘𝑋))})) |
257 | | df-ima 5602 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {(𝑋𝐹(𝑊‘𝑋))}) = ran (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) |
258 | | cnvxp 6060 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
259 | 258 | reseq1i 5887 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = (({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) ↾ {(𝑋𝐹(𝑊‘𝑋))}) |
260 | | ssid 3943 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ {(𝑋𝐹(𝑊‘𝑋))} ⊆ {(𝑋𝐹(𝑊‘𝑋))} |
261 | | xpssres 5928 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ⊆ {(𝑋𝐹(𝑊‘𝑋))} → (({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋)) |
262 | 260, 261 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
263 | 259, 262 | eqtri 2766 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
264 | 263 | rneqi 5846 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran
(◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = ran ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) |
265 | 46 | snnz 4712 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ {(𝑋𝐹(𝑊‘𝑋))} ≠ ∅ |
266 | | rnxp 6073 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ≠ ∅ → ran ({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) = 𝑋) |
267 | 265, 266 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran
({(𝑋𝐹(𝑊‘𝑋))} × 𝑋) = 𝑋 |
268 | 264, 267 | eqtri 2766 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ran
(◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ↾ {(𝑋𝐹(𝑊‘𝑋))}) = 𝑋 |
269 | 257, 268 | eqtri 2766 |
. . . . . . . . . . . . . . . . . . 19
⊢ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {(𝑋𝐹(𝑊‘𝑋))}) = 𝑋 |
270 | 256, 269 | eqtrdi 2794 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦}) = 𝑋) |
271 | 254, 270 | uneq12d 4098 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((◡(𝑊‘𝑋) “ {𝑦}) ∪ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦})) = (∅ ∪ 𝑋)) |
272 | | cnvun 6046 |
. . . . . . . . . . . . . . . . . . 19
⊢ ◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) = (◡(𝑊‘𝑋) ∪ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) |
273 | 272 | imaeq1i 5966 |
. . . . . . . . . . . . . . . . . 18
⊢ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = ((◡(𝑊‘𝑋) ∪ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) |
274 | | imaundir 6054 |
. . . . . . . . . . . . . . . . . 18
⊢ ((◡(𝑊‘𝑋) ∪ ◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = ((◡(𝑊‘𝑋) “ {𝑦}) ∪ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦})) |
275 | 273, 274 | eqtri 2766 |
. . . . . . . . . . . . . . . . 17
⊢ (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = ((◡(𝑊‘𝑋) “ {𝑦}) ∪ (◡(𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) “ {𝑦})) |
276 | | un0 4324 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∪ ∅) = 𝑋 |
277 | | uncom 4087 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∪ ∅) = (∅ ∪
𝑋) |
278 | 276, 277 | eqtr3i 2768 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑋 = (∅ ∪ 𝑋) |
279 | 271, 275,
278 | 3eqtr4g 2803 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) = 𝑋) |
280 | 189, 279 | sylan9eqr 2800 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → 𝑢 = 𝑋) |
281 | 280 | sqxpeqd 5621 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢 × 𝑢) = (𝑋 × 𝑋)) |
282 | 281 | ineq2d 4146 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋))) |
283 | | indir 4209 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = (((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋))) |
284 | | df-ss 3904 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) ↔ ((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
285 | 244, 284 | sylib 217 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
286 | | incom 4135 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋) = (𝑋 ∩ {(𝑋𝐹(𝑊‘𝑋))}) |
287 | | disjsn 4647 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑋 ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅ ↔ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
288 | 242, 287 | sylibr 233 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑋 ∩ {(𝑋𝐹(𝑊‘𝑋))}) = ∅) |
289 | 286, 288 | eqtrid 2790 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋) = ∅) |
290 | 289 | xpeq2d 5619 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (𝑋 × ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋)) = (𝑋 × ∅)) |
291 | | xpindi 5742 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑋 × ({(𝑋𝐹(𝑊‘𝑋))} ∩ 𝑋)) = ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋)) |
292 | | xp0 6061 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑋 × ∅) =
∅ |
293 | 290, 291,
292 | 3eqtr3g 2801 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋)) = ∅) |
294 | 285, 293 | uneq12d 4098 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (((𝑊‘𝑋) ∩ (𝑋 × 𝑋)) ∪ ((𝑋 × {(𝑋𝐹(𝑊‘𝑋))}) ∩ (𝑋 × 𝑋))) = ((𝑊‘𝑋) ∪ ∅)) |
295 | 283, 294 | eqtrid 2790 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = ((𝑊‘𝑋) ∪ ∅)) |
296 | | un0 4324 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊‘𝑋) ∪ ∅) = (𝑊‘𝑋) |
297 | 295, 296 | eqtrdi 2794 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
298 | 297 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑋 × 𝑋)) = (𝑊‘𝑋)) |
299 | 282, 298 | eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢)) = (𝑊‘𝑋)) |
300 | 280, 299 | oveq12d 7293 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → (𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = (𝑋𝐹(𝑊‘𝑋))) |
301 | 300 | eqeq1d 2740 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) ∧ 𝑢 = (◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦})) → ((𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑋𝐹(𝑊‘𝑋)) = 𝑦)) |
302 | 241, 301 | sbcied 3761 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → ([(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑋𝐹(𝑊‘𝑋)) = 𝑦)) |
303 | 240, 302 | mpbird 256 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))}) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
304 | 238, 303 | jaodan 955 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∨ 𝑦 ∈ {(𝑋𝐹(𝑊‘𝑋))})) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
305 | 135, 304 | sylan2b 594 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) ∧ 𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})) → [(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
306 | 305 | ralrimiva 3103 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦) |
307 | 175, 306 | jca 512 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦)) |
308 | 2, 3 | fpwwe2lem2 10388 |
. . . . . . . 8
⊢ (𝜑 → ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ↔ (((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴 ∧ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) ∧ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦)))) |
309 | 308 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ↔ (((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝐴 ∧ ((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ⊆ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) × (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}))) ∧ (((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) We (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∧ ∀𝑦 ∈ (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})[(◡((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) “ {𝑦}) / 𝑢](𝑢𝐹(((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) ∩ (𝑢 × 𝑢))) = 𝑦)))) |
310 | 34, 307, 309 | mpbir2and 710 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))}))) |
311 | 2 | relopabiv 5730 |
. . . . . . 7
⊢ Rel 𝑊 |
312 | 311 | releldmi 5857 |
. . . . . 6
⊢ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))})𝑊((𝑊‘𝑋) ∪ (𝑋 × {(𝑋𝐹(𝑊‘𝑋))})) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∈ dom 𝑊) |
313 | | elssuni 4871 |
. . . . . 6
⊢ ((𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ∈ dom 𝑊 → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ∪
dom 𝑊) |
314 | 310, 312,
313 | 3syl 18 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ ∪
dom 𝑊) |
315 | 314, 7 | sseqtrrdi 3972 |
. . . 4
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋 ∪ {(𝑋𝐹(𝑊‘𝑋))}) ⊆ 𝑋) |
316 | 1, 315 | sstrid 3932 |
. . 3
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → {(𝑋𝐹(𝑊‘𝑋))} ⊆ 𝑋) |
317 | 46 | snss 4719 |
. . 3
⊢ ((𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋 ↔ {(𝑋𝐹(𝑊‘𝑋))} ⊆ 𝑋) |
318 | 316, 317 | sylibr 233 |
. 2
⊢ ((𝜑 ∧ ¬ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
319 | 318 | pm2.18da 797 |
1
⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |