| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | vex 3483 | . . 3
⊢ 𝑧 ∈ V | 
| 2 | 1 | eldm2 5911 | . 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 8316 | . . . . . . 7
⊢ 𝐹 = ∪
𝐵 | 
| 6 | 3 | frrlem1 8312 | . . . . . . . 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 4918 | . . . . . . 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 2764 | . . . . . 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 2832 | . . . . 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 4920 | . . . . 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 1233 | . . . . . . . . . . 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 3483 | . . . . . . . . . . . . . 14
⊢ 𝑤 ∈ V | 
| 14 | 1, 13 | opeldm 5917 | . . . . . . . . . . . . 13
⊢
(〈𝑧, 𝑤〉 ∈ 𝑔 → 𝑧 ∈ dom 𝑔) | 
| 15 | 14 | adantr 480 | . . . . . . . . . . . 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 1194 | . . . . . . . . . . . . 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 6672 | . . . . . . . . . . . 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 2842 | . . . . . . . . . . 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 3246 | . . . . . . . . . . 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 4020 | . . . . . . . . 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 2180 | . . . . . . . . . . . . . 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 | eqabri 2884 | . . . . . . . . . . . . . 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 234 | . . . . . . . . . . . . 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 481 | . . . . . . . . . . . 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 4936 | . . . . . . . . . . . 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 4024 | . . . . . . . . . 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 5912 | . . . . . . . . . 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 3993 | . . . . . . . 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 413 | . . . . . . 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 1929 | . . . . . 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 407 | . . . . 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 1929 | . . . 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 217 | . . 3
⊢
(〈𝑧, 𝑤〉 ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) | 
| 37 | 36 | exlimiv 1929 | . 2
⊢
(∃𝑤〈𝑧, 𝑤〉 ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) | 
| 38 | 2, 37 | sylbi 217 | 1
⊢ (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |