| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpll 766 | . . . . 5
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈
1stω) | 
| 2 |  | 1stctop 23452 | . . . . . . 7
⊢ (𝐽 ∈ 1stω
→ 𝐽 ∈
Top) | 
| 3 |  | 1stcelcls.1 | . . . . . . . 8
⊢ 𝑋 = ∪
𝐽 | 
| 4 | 3 | clsss3 23068 | . . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) | 
| 5 | 2, 4 | sylan 580 | . . . . . 6
⊢ ((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) | 
| 6 | 5 | sselda 3982 | . . . . 5
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ 𝑋) | 
| 7 | 3 | 1stcfb 23454 | . . . . 5
⊢ ((𝐽 ∈ 1stω
∧ 𝑃 ∈ 𝑋) → ∃𝑔(𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) | 
| 8 | 1, 6, 7 | syl2anc 584 | . . . 4
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑔(𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) | 
| 9 |  | simpr2 1195 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘))) | 
| 10 |  | simpl 482 | . . . . . . . . . . . . 13
⊢ ((𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → 𝑃 ∈ (𝑔‘𝑘)) | 
| 11 | 10 | ralimi 3082 | . . . . . . . . . . . 12
⊢
(∀𝑘 ∈
ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → ∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔‘𝑘)) | 
| 12 | 9, 11 | syl 17 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔‘𝑘)) | 
| 13 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛)) | 
| 14 | 13 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ (𝑘 = 𝑛 → (𝑃 ∈ (𝑔‘𝑘) ↔ 𝑃 ∈ (𝑔‘𝑛))) | 
| 15 | 14 | rspccva 3620 | . . . . . . . . . . 11
⊢
((∀𝑘 ∈
ℕ 𝑃 ∈ (𝑔‘𝑘) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ (𝑔‘𝑛)) | 
| 16 | 12, 15 | sylan 580 | . . . . . . . . . 10
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ (𝑔‘𝑛)) | 
| 17 |  | eleq2 2829 | . . . . . . . . . . . 12
⊢ (𝑦 = (𝑔‘𝑛) → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ (𝑔‘𝑛))) | 
| 18 |  | ineq1 4212 | . . . . . . . . . . . . 13
⊢ (𝑦 = (𝑔‘𝑛) → (𝑦 ∩ 𝑆) = ((𝑔‘𝑛) ∩ 𝑆)) | 
| 19 | 18 | neeq1d 2999 | . . . . . . . . . . . 12
⊢ (𝑦 = (𝑔‘𝑛) → ((𝑦 ∩ 𝑆) ≠ ∅ ↔ ((𝑔‘𝑛) ∩ 𝑆) ≠ ∅)) | 
| 20 | 17, 19 | imbi12d 344 | . . . . . . . . . . 11
⊢ (𝑦 = (𝑔‘𝑛) → ((𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑔‘𝑛) → ((𝑔‘𝑛) ∩ 𝑆) ≠ ∅))) | 
| 21 | 3 | elcls2 23083 | . . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅)))) | 
| 22 | 2, 21 | sylan 580 | . . . . . . . . . . . . 13
⊢ ((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅)))) | 
| 23 | 22 | simplbda 499 | . . . . . . . . . . . 12
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅)) | 
| 24 | 23 | ad2antrr 726 | . . . . . . . . . . 11
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅)) | 
| 25 |  | simpr1 1194 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → 𝑔:ℕ⟶𝐽) | 
| 26 | 25 | ffvelcdmda 7103 | . . . . . . . . . . 11
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ 𝐽) | 
| 27 | 20, 24, 26 | rspcdva 3622 | . . . . . . . . . 10
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → (𝑃 ∈ (𝑔‘𝑛) → ((𝑔‘𝑛) ∩ 𝑆) ≠ ∅)) | 
| 28 | 16, 27 | mpd 15 | . . . . . . . . 9
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ((𝑔‘𝑛) ∩ 𝑆) ≠ ∅) | 
| 29 |  | elin 3966 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆) ↔ (𝑥 ∈ (𝑔‘𝑛) ∧ 𝑥 ∈ 𝑆)) | 
| 30 | 29 | biancomi 462 | . . . . . . . . . . 11
⊢ (𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑔‘𝑛))) | 
| 31 | 30 | exbii 1847 | . . . . . . . . . 10
⊢
(∃𝑥 𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑔‘𝑛))) | 
| 32 |  | n0 4352 | . . . . . . . . . 10
⊢ (((𝑔‘𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆)) | 
| 33 |  | df-rex 3070 | . . . . . . . . . 10
⊢
(∃𝑥 ∈
𝑆 𝑥 ∈ (𝑔‘𝑛) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑔‘𝑛))) | 
| 34 | 31, 32, 33 | 3bitr4i 303 | . . . . . . . . 9
⊢ (((𝑔‘𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑔‘𝑛)) | 
| 35 | 28, 34 | sylib 218 | . . . . . . . 8
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑔‘𝑛)) | 
| 36 | 2 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top) | 
| 37 | 3 | topopn 22913 | . . . . . . . . . . . 12
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) | 
| 38 | 36, 37 | syl 17 | . . . . . . . . . . 11
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 ∈ 𝐽) | 
| 39 |  | simplr 768 | . . . . . . . . . . 11
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) | 
| 40 | 38, 39 | ssexd 5323 | . . . . . . . . . 10
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V) | 
| 41 |  | fvi 6984 | . . . . . . . . . 10
⊢ (𝑆 ∈ V → ( I
‘𝑆) = 𝑆) | 
| 42 | 40, 41 | syl 17 | . . . . . . . . 9
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ( I ‘𝑆) = 𝑆) | 
| 43 | 42 | ad2antrr 726 | . . . . . . . 8
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ( I ‘𝑆) = 𝑆) | 
| 44 | 35, 43 | rexeqtrrdv 3330 | . . . . . . 7
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔‘𝑛)) | 
| 45 | 44 | ralrimiva 3145 | . . . . . 6
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ∀𝑛 ∈ ℕ ∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔‘𝑛)) | 
| 46 |  | fvex 6918 | . . . . . . 7
⊢ ( I
‘𝑆) ∈
V | 
| 47 |  | nnenom 14022 | . . . . . . 7
⊢ ℕ
≈ ω | 
| 48 |  | eleq1 2828 | . . . . . . 7
⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ∈ (𝑔‘𝑛) ↔ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) | 
| 49 | 46, 47, 48 | axcc4 10480 | . . . . . 6
⊢
(∀𝑛 ∈
ℕ ∃𝑥 ∈ ( I
‘𝑆)𝑥 ∈ (𝑔‘𝑛) → ∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) | 
| 50 | 45, 49 | syl 17 | . . . . 5
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) | 
| 51 | 42 | feq3d 6722 | . . . . . . . . 9
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) ↔ 𝑓:ℕ⟶𝑆)) | 
| 52 | 51 | biimpd 229 | . . . . . . . 8
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) → 𝑓:ℕ⟶𝑆)) | 
| 53 | 52 | adantr 480 | . . . . . . 7
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → (𝑓:ℕ⟶( I ‘𝑆) → 𝑓:ℕ⟶𝑆)) | 
| 54 | 6 | ad2antrr 726 | . . . . . . . . . 10
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 𝑃 ∈ 𝑋) | 
| 55 |  | simplr3 1217 | . . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥)) | 
| 56 |  | eleq2 2829 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦)) | 
| 57 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑗 → (𝑔‘𝑘) = (𝑔‘𝑗)) | 
| 58 | 57 | sseq1d 4014 | . . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → ((𝑔‘𝑘) ⊆ 𝑥 ↔ (𝑔‘𝑗) ⊆ 𝑥)) | 
| 59 | 58 | cbvrexvw 3237 | . . . . . . . . . . . . . . . 16
⊢
(∃𝑘 ∈
ℕ (𝑔‘𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑥) | 
| 60 |  | sseq2 4009 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → ((𝑔‘𝑗) ⊆ 𝑥 ↔ (𝑔‘𝑗) ⊆ 𝑦)) | 
| 61 | 60 | rexbidv 3178 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦)) | 
| 62 | 59, 61 | bitrid 283 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦)) | 
| 63 | 56, 62 | imbi12d 344 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥) ↔ (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦))) | 
| 64 | 63 | rspccva 3620 | . . . . . . . . . . . . 13
⊢
((∀𝑥 ∈
𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥) ∧ 𝑦 ∈ 𝐽) → (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦)) | 
| 65 | 55, 64 | sylan 580 | . . . . . . . . . . . 12
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) ∧ 𝑦 ∈ 𝐽) → (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦)) | 
| 66 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) | 
| 67 | 66 | ralimi 3082 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑘 ∈
ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) | 
| 68 | 9, 67 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) | 
| 69 | 68 | adantr 480 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) | 
| 70 |  | simprrr 781 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) → 𝑗 ∈ ℕ) | 
| 71 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑗 → (𝑔‘𝑛) = (𝑔‘𝑗)) | 
| 72 | 71 | sseq1d 4014 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑗 → ((𝑔‘𝑛) ⊆ (𝑔‘𝑗) ↔ (𝑔‘𝑗) ⊆ (𝑔‘𝑗))) | 
| 73 | 72 | imbi2d 340 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑗 → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑛) ⊆ (𝑔‘𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑗) ⊆ (𝑔‘𝑗)))) | 
| 74 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚)) | 
| 75 | 74 | sseq1d 4014 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑔‘𝑛) ⊆ (𝑔‘𝑗) ↔ (𝑔‘𝑚) ⊆ (𝑔‘𝑗))) | 
| 76 | 75 | imbi2d 340 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑛) ⊆ (𝑔‘𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑚) ⊆ (𝑔‘𝑗)))) | 
| 77 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = (𝑚 + 1) → (𝑔‘𝑛) = (𝑔‘(𝑚 + 1))) | 
| 78 | 77 | sseq1d 4014 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = (𝑚 + 1) → ((𝑔‘𝑛) ⊆ (𝑔‘𝑗) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗))) | 
| 79 | 78 | imbi2d 340 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = (𝑚 + 1) → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑛) ⊆ (𝑔‘𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗)))) | 
| 80 |  | ssid 4005 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑔‘𝑗) ⊆ (𝑔‘𝑗) | 
| 81 | 80 | 2a1i 12 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ ℤ →
((∀𝑘 ∈ ℕ
(𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑗) ⊆ (𝑔‘𝑗))) | 
| 82 |  | eluznn 12961 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑗 ∈ ℕ ∧ 𝑚 ∈
(ℤ≥‘𝑗)) → 𝑚 ∈ ℕ) | 
| 83 |  | fvoveq1 7455 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1))) | 
| 84 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 = 𝑚 → (𝑔‘𝑘) = (𝑔‘𝑚)) | 
| 85 | 83, 84 | sseq12d 4016 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚))) | 
| 86 | 85 | rspccva 3620 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((∀𝑘 ∈
ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚)) | 
| 87 | 82, 86 | sylan2 593 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((∀𝑘 ∈
ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ (𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ≥‘𝑗))) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚)) | 
| 88 | 87 | anassrs 467 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((∀𝑘 ∈
ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚)) | 
| 89 |  | sstr2 3989 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚) → ((𝑔‘𝑚) ⊆ (𝑔‘𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗))) | 
| 90 | 88, 89 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((∀𝑘 ∈
ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ((𝑔‘𝑚) ⊆ (𝑔‘𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗))) | 
| 91 | 90 | expcom 413 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈
(ℤ≥‘𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → ((𝑔‘𝑚) ⊆ (𝑔‘𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗)))) | 
| 92 | 91 | a2d 29 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈
(ℤ≥‘𝑗) → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑚) ⊆ (𝑔‘𝑗)) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗)))) | 
| 93 | 73, 76, 79, 76, 81, 92 | uzind4 12949 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈
(ℤ≥‘𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑚) ⊆ (𝑔‘𝑗))) | 
| 94 | 93 | com12 32 | . . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑘 ∈
ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑚 ∈ (ℤ≥‘𝑗) → (𝑔‘𝑚) ⊆ (𝑔‘𝑗))) | 
| 95 | 94 | ralrimiv 3144 | . . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑘 ∈
ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → ∀𝑚 ∈
(ℤ≥‘𝑗)(𝑔‘𝑚) ⊆ (𝑔‘𝑗)) | 
| 96 | 69, 70, 95 | syl2anc 584 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) → ∀𝑚 ∈
(ℤ≥‘𝑗)(𝑔‘𝑚) ⊆ (𝑔‘𝑗)) | 
| 97 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚)) | 
| 98 | 97, 74 | eleq12d 2834 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 𝑚 → ((𝑓‘𝑛) ∈ (𝑔‘𝑛) ↔ (𝑓‘𝑚) ∈ (𝑔‘𝑚))) | 
| 99 |  | simplr 768 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) | 
| 100 | 99 | ad2antlr 727 | . . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) | 
| 101 | 70, 82 | sylan 580 | . . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℕ) | 
| 102 | 98, 100, 101 | rspcdva 3622 | . . . . . . . . . . . . . . . . . . 19
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝑓‘𝑚) ∈ (𝑔‘𝑚)) | 
| 103 | 102 | ralrimiva 3145 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) → ∀𝑚 ∈
(ℤ≥‘𝑗)(𝑓‘𝑚) ∈ (𝑔‘𝑚)) | 
| 104 |  | r19.26 3110 | . . . . . . . . . . . . . . . . . 18
⊢
(∀𝑚 ∈
(ℤ≥‘𝑗)((𝑔‘𝑚) ⊆ (𝑔‘𝑗) ∧ (𝑓‘𝑚) ∈ (𝑔‘𝑚)) ↔ (∀𝑚 ∈ (ℤ≥‘𝑗)(𝑔‘𝑚) ⊆ (𝑔‘𝑗) ∧ ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ (𝑔‘𝑚))) | 
| 105 | 96, 103, 104 | sylanbrc 583 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) → ∀𝑚 ∈
(ℤ≥‘𝑗)((𝑔‘𝑚) ⊆ (𝑔‘𝑗) ∧ (𝑓‘𝑚) ∈ (𝑔‘𝑚))) | 
| 106 |  | ssel2 3977 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝑔‘𝑚) ⊆ (𝑔‘𝑗) ∧ (𝑓‘𝑚) ∈ (𝑔‘𝑚)) → (𝑓‘𝑚) ∈ (𝑔‘𝑗)) | 
| 107 | 106 | ralimi 3082 | . . . . . . . . . . . . . . . . 17
⊢
(∀𝑚 ∈
(ℤ≥‘𝑗)((𝑔‘𝑚) ⊆ (𝑔‘𝑗) ∧ (𝑓‘𝑚) ∈ (𝑔‘𝑚)) → ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ (𝑔‘𝑗)) | 
| 108 | 105, 107 | syl 17 | . . . . . . . . . . . . . . . 16
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) → ∀𝑚 ∈
(ℤ≥‘𝑗)(𝑓‘𝑚) ∈ (𝑔‘𝑗)) | 
| 109 |  | ssel 3976 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑔‘𝑗) ⊆ 𝑦 → ((𝑓‘𝑚) ∈ (𝑔‘𝑗) → (𝑓‘𝑚) ∈ 𝑦)) | 
| 110 | 109 | ralimdv 3168 | . . . . . . . . . . . . . . . 16
⊢ ((𝑔‘𝑗) ⊆ 𝑦 → (∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ (𝑔‘𝑗) → ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)) | 
| 111 | 108, 110 | syl5com 31 | . . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ))) → ((𝑔‘𝑗) ⊆ 𝑦 → ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)) | 
| 112 | 111 | anassrs 467 | . . . . . . . . . . . . . 14
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ)) → ((𝑔‘𝑗) ⊆ 𝑦 → ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)) | 
| 113 | 112 | anassrs 467 | . . . . . . . . . . . . 13
⊢
(((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) ∧ 𝑦 ∈ 𝐽) ∧ 𝑗 ∈ ℕ) → ((𝑔‘𝑗) ⊆ 𝑦 → ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)) | 
| 114 | 113 | reximdva 3167 | . . . . . . . . . . . 12
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) ∧ 𝑦 ∈ 𝐽) → (∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)) | 
| 115 | 65, 114 | syld 47 | . . . . . . . . . . 11
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) ∧ 𝑦 ∈ 𝐽) → (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)) | 
| 116 | 115 | ralrimiva 3145 | . . . . . . . . . 10
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)) | 
| 117 | 36 | ad2antrr 726 | . . . . . . . . . . . 12
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 𝐽 ∈ Top) | 
| 118 | 3 | toptopon 22924 | . . . . . . . . . . . 12
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) | 
| 119 | 117, 118 | sylib 218 | . . . . . . . . . . 11
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 120 |  | nnuz 12922 | . . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) | 
| 121 |  | 1zzd 12650 | . . . . . . . . . . 11
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 1 ∈ ℤ) | 
| 122 |  | simprl 770 | . . . . . . . . . . . 12
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 𝑓:ℕ⟶𝑆) | 
| 123 | 39 | ad2antrr 726 | . . . . . . . . . . . 12
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 𝑆 ⊆ 𝑋) | 
| 124 | 122, 123 | fssd 6752 | . . . . . . . . . . 11
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 𝑓:ℕ⟶𝑋) | 
| 125 |  | eqidd 2737 | . . . . . . . . . . 11
⊢
((((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) = (𝑓‘𝑚)) | 
| 126 | 119, 120,
121, 124, 125 | lmbrf 23269 | . . . . . . . . . 10
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → (𝑓(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦)))) | 
| 127 | 54, 116, 126 | mpbir2and 713 | . . . . . . . . 9
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))) → 𝑓(⇝𝑡‘𝐽)𝑃) | 
| 128 | 127 | expr 456 | . . . . . . . 8
⊢
(((((𝐽 ∈
1stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) ∧ 𝑓:ℕ⟶𝑆) → (∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛) → 𝑓(⇝𝑡‘𝐽)𝑃)) | 
| 129 | 128 | imdistanda 571 | . . . . . . 7
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) → (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) | 
| 130 | 53, 129 | syland 603 | . . . . . 6
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ((𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) → (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) | 
| 131 | 130 | eximdv 1916 | . . . . 5
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → (∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) | 
| 132 | 50, 131 | mpd 15 | . . . 4
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥))) → ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) | 
| 133 | 8, 132 | exlimddv 1934 | . . 3
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) | 
| 134 | 133 | ex 412 | . 2
⊢ ((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) | 
| 135 | 2 | ad2antrr 726 | . . . . . 6
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) → 𝐽 ∈ Top) | 
| 136 | 135, 118 | sylib 218 | . . . . 5
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 137 |  | 1zzd 12650 | . . . . 5
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) → 1 ∈ ℤ) | 
| 138 |  | simprr 772 | . . . . 5
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) → 𝑓(⇝𝑡‘𝐽)𝑃) | 
| 139 |  | simprl 770 | . . . . . 6
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) → 𝑓:ℕ⟶𝑆) | 
| 140 | 139 | ffvelcdmda 7103 | . . . . 5
⊢ ((((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝑆) | 
| 141 |  | simplr 768 | . . . . 5
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) → 𝑆 ⊆ 𝑋) | 
| 142 | 120, 136,
137, 138, 140, 141 | lmcls 23311 | . . . 4
⊢ (((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃)) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) | 
| 143 | 142 | ex 412 | . . 3
⊢ ((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) → ((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆))) | 
| 144 | 143 | exlimdv 1932 | . 2
⊢ ((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) → (∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆))) | 
| 145 | 134, 144 | impbid 212 | 1
⊢ ((𝐽 ∈ 1stω
∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) |