Step | Hyp | Ref
| Expression |
1 | | vex 3451 |
. . 3
⊢ 𝑧 ∈ V |
2 | 1 | eldm2 5861 |
. 2
⊢ (𝑧 ∈ dom 𝐹 ↔ ∃𝑤⟨𝑧, 𝑤⟩ ∈ 𝐹) |
3 | | frrlem5.1 |
. . . . . . . 8
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
4 | | frrlem5.2 |
. . . . . . . 8
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
5 | 3, 4 | frrlem5 8225 |
. . . . . . 7
⊢ 𝐹 = ∪
𝐵 |
6 | 3 | frrlem1 8221 |
. . . . . . . 8
⊢ 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))} |
7 | 6 | unieqi 4882 |
. . . . . . 7
⊢ ∪ 𝐵 =
∪ {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))} |
8 | 5, 7 | eqtri 2761 |
. . . . . 6
⊢ 𝐹 = ∪
{𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))} |
9 | 8 | eleq2i 2826 |
. . . . 5
⊢
(⟨𝑧, 𝑤⟩ ∈ 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ∪
{𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}) |
10 | | eluniab 4884 |
. . . . 5
⊢
(⟨𝑧, 𝑤⟩ ∈ ∪ {𝑔
∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))} ↔ ∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))) |
11 | 9, 10 | bitri 275 |
. . . 4
⊢
(⟨𝑧, 𝑤⟩ ∈ 𝐹 ↔ ∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))) |
12 | | simpr2r 1234 |
. . . . . . . . . . 11
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) |
13 | | vex 3451 |
. . . . . . . . . . . . . 14
⊢ 𝑤 ∈ V |
14 | 1, 13 | opeldm 5867 |
. . . . . . . . . . . . 13
⊢
(⟨𝑧, 𝑤⟩ ∈ 𝑔 → 𝑧 ∈ dom 𝑔) |
15 | 14 | adantr 482 |
. . . . . . . . . . . 12
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑧 ∈ dom 𝑔) |
16 | | simpr1 1195 |
. . . . . . . . . . . . 13
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 Fn 𝑎) |
17 | 16 | fndmd 6611 |
. . . . . . . . . . . 12
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → dom 𝑔 = 𝑎) |
18 | 15, 17 | eleqtrd 2836 |
. . . . . . . . . . 11
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑧 ∈ 𝑎) |
19 | | rsp 3229 |
. . . . . . . . . . 11
⊢
(∀𝑧 ∈
𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎 → (𝑧 ∈ 𝑎 → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎)) |
20 | 12, 18, 19 | sylc 65 |
. . . . . . . . . 10
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) |
21 | 20, 17 | sseqtrrd 3989 |
. . . . . . . . 9
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝑔) |
22 | | 19.8a 2175 |
. . . . . . . . . . . . . 14
⊢ ((𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) |
23 | 6 | eqabi 2878 |
. . . . . . . . . . . . . 14
⊢ (𝑔 ∈ 𝐵 ↔ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) |
24 | 22, 23 | sylibr 233 |
. . . . . . . . . . . . 13
⊢ ((𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝑔 ∈ 𝐵) |
25 | 24 | adantl 483 |
. . . . . . . . . . . 12
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 ∈ 𝐵) |
26 | | elssuni 4902 |
. . . . . . . . . . . 12
⊢ (𝑔 ∈ 𝐵 → 𝑔 ⊆ ∪ 𝐵) |
27 | 25, 26 | syl 17 |
. . . . . . . . . . 11
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 ⊆ ∪ 𝐵) |
28 | 27, 5 | sseqtrrdi 3999 |
. . . . . . . . . 10
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 ⊆ 𝐹) |
29 | | dmss 5862 |
. . . . . . . . . 10
⊢ (𝑔 ⊆ 𝐹 → dom 𝑔 ⊆ dom 𝐹) |
30 | 28, 29 | syl 17 |
. . . . . . . . 9
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → dom 𝑔 ⊆ dom 𝐹) |
31 | 21, 30 | sstrd 3958 |
. . . . . . . 8
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
32 | 31 | expcom 415 |
. . . . . . 7
⊢ ((𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)) |
33 | 32 | exlimiv 1934 |
. . . . . 6
⊢
(∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)) |
34 | 33 | impcom 409 |
. . . . 5
⊢
((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
35 | 34 | exlimiv 1934 |
. . . 4
⊢
(∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
36 | 11, 35 | sylbi 216 |
. . 3
⊢
(⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
37 | 36 | exlimiv 1934 |
. 2
⊢
(∃𝑤⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
38 | 2, 37 | sylbi 216 |
1
⊢ (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |