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Theorem 1stcelcls 23470
Description: A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 10476. A space satisfying the conclusion of this theorem is called a sequential space, so the theorem can also be stated as "every first-countable space is a sequential space". (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
1stcelcls.1 𝑋 = 𝐽
Assertion
Ref Expression
1stcelcls ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
Distinct variable groups:   𝑓,𝐽   𝑃,𝑓   𝑆,𝑓   𝑓,𝑋

Proof of Theorem 1stcelcls
Dummy variables 𝑔 𝑗 𝑘 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 766 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ 1stω)
2 1stctop 23452 . . . . . . 7 (𝐽 ∈ 1stω → 𝐽 ∈ Top)
3 1stcelcls.1 . . . . . . . 8 𝑋 = 𝐽
43clsss3 23068 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
52, 4sylan 580 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
65sselda 3982 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃𝑋)
731stcfb 23454 . . . . 5 ((𝐽 ∈ 1stω ∧ 𝑃𝑋) → ∃𝑔(𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥)))
81, 6, 7syl2anc 584 . . . 4 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑔(𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥)))
9 simpr2 1195 . . . . . . . . . . . 12 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)))
10 simpl 482 . . . . . . . . . . . . 13 ((𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → 𝑃 ∈ (𝑔𝑘))
1110ralimi 3082 . . . . . . . . . . . 12 (∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → ∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔𝑘))
129, 11syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔𝑘))
13 fveq2 6905 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑔𝑘) = (𝑔𝑛))
1413eleq2d 2826 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑃 ∈ (𝑔𝑘) ↔ 𝑃 ∈ (𝑔𝑛)))
1514rspccva 3620 . . . . . . . . . . 11 ((∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔𝑘) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ (𝑔𝑛))
1612, 15sylan 580 . . . . . . . . . 10 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ (𝑔𝑛))
17 eleq2 2829 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → (𝑃𝑦𝑃 ∈ (𝑔𝑛)))
18 ineq1 4212 . . . . . . . . . . . . 13 (𝑦 = (𝑔𝑛) → (𝑦𝑆) = ((𝑔𝑛) ∩ 𝑆))
1918neeq1d 2999 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → ((𝑦𝑆) ≠ ∅ ↔ ((𝑔𝑛) ∩ 𝑆) ≠ ∅))
2017, 19imbi12d 344 . . . . . . . . . . 11 (𝑦 = (𝑔𝑛) → ((𝑃𝑦 → (𝑦𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑔𝑛) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅)))
213elcls2 23083 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))))
222, 21sylan 580 . . . . . . . . . . . . 13 ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))))
2322simplbda 499 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))
2423ad2antrr 726 . . . . . . . . . . 11 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))
25 simpr1 1194 . . . . . . . . . . . 12 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → 𝑔:ℕ⟶𝐽)
2625ffvelcdmda 7103 . . . . . . . . . . 11 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → (𝑔𝑛) ∈ 𝐽)
2720, 24, 26rspcdva 3622 . . . . . . . . . 10 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → (𝑃 ∈ (𝑔𝑛) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅))
2816, 27mpd 15 . . . . . . . . 9 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅)
29 elin 3966 . . . . . . . . . . . 12 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥 ∈ (𝑔𝑛) ∧ 𝑥𝑆))
3029biancomi 462 . . . . . . . . . . 11 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3130exbii 1847 . . . . . . . . . 10 (∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ ∃𝑥(𝑥𝑆𝑥 ∈ (𝑔𝑛)))
32 n0 4352 . . . . . . . . . 10 (((𝑔𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆))
33 df-rex 3070 . . . . . . . . . 10 (∃𝑥𝑆 𝑥 ∈ (𝑔𝑛) ↔ ∃𝑥(𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3431, 32, 333bitr4i 303 . . . . . . . . 9 (((𝑔𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥𝑆 𝑥 ∈ (𝑔𝑛))
3528, 34sylib 218 . . . . . . . 8 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ∃𝑥𝑆 𝑥 ∈ (𝑔𝑛))
362ad2antrr 726 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
373topopn 22913 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝑋𝐽)
3836, 37syl 17 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋𝐽)
39 simplr 768 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
4038, 39ssexd 5323 . . . . . . . . . 10 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
41 fvi 6984 . . . . . . . . . 10 (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
4240, 41syl 17 . . . . . . . . 9 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ( I ‘𝑆) = 𝑆)
4342ad2antrr 726 . . . . . . . 8 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ( I ‘𝑆) = 𝑆)
4435, 43rexeqtrrdv 3330 . . . . . . 7 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔𝑛))
4544ralrimiva 3145 . . . . . 6 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ∀𝑛 ∈ ℕ ∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔𝑛))
46 fvex 6918 . . . . . . 7 ( I ‘𝑆) ∈ V
47 nnenom 14022 . . . . . . 7 ℕ ≈ ω
48 eleq1 2828 . . . . . . 7 (𝑥 = (𝑓𝑛) → (𝑥 ∈ (𝑔𝑛) ↔ (𝑓𝑛) ∈ (𝑔𝑛)))
4946, 47, 48axcc4 10480 . . . . . 6 (∀𝑛 ∈ ℕ ∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔𝑛) → ∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)))
5045, 49syl 17 . . . . 5 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)))
5142feq3d 6722 . . . . . . . . 9 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) ↔ 𝑓:ℕ⟶𝑆))
5251biimpd 229 . . . . . . . 8 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) → 𝑓:ℕ⟶𝑆))
5352adantr 480 . . . . . . 7 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → (𝑓:ℕ⟶( I ‘𝑆) → 𝑓:ℕ⟶𝑆))
546ad2antrr 726 . . . . . . . . . 10 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 𝑃𝑋)
55 simplr3 1217 . . . . . . . . . . . . 13 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))
56 eleq2 2829 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
57 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → (𝑔𝑘) = (𝑔𝑗))
5857sseq1d 4014 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → ((𝑔𝑘) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑥))
5958cbvrexvw 3237 . . . . . . . . . . . . . . . 16 (∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑥)
60 sseq2 4009 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((𝑔𝑗) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑦))
6160rexbidv 3178 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6259, 61bitrid 283 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6356, 62imbi12d 344 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥) ↔ (𝑃𝑦 → ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦)))
6463rspccva 3620 . . . . . . . . . . . . 13 ((∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥) ∧ 𝑦𝐽) → (𝑃𝑦 → ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6555, 64sylan 580 . . . . . . . . . . . 12 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) ∧ 𝑦𝐽) → (𝑃𝑦 → ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
66 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘))
6766ralimi 3082 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘))
689, 67syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘))
6968adantr 480 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘))
70 simprrr 781 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) → 𝑗 ∈ ℕ)
71 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑗 → (𝑔𝑛) = (𝑔𝑗))
7271sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑗 → ((𝑔𝑛) ⊆ (𝑔𝑗) ↔ (𝑔𝑗) ⊆ (𝑔𝑗)))
7372imbi2d 340 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑛) ⊆ (𝑔𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑗) ⊆ (𝑔𝑗))))
74 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
7574sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑔𝑛) ⊆ (𝑔𝑗) ↔ (𝑔𝑚) ⊆ (𝑔𝑗)))
7675imbi2d 340 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑛) ⊆ (𝑔𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑚) ⊆ (𝑔𝑗))))
77 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 + 1) → (𝑔𝑛) = (𝑔‘(𝑚 + 1)))
7877sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = (𝑚 + 1) → ((𝑔𝑛) ⊆ (𝑔𝑗) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗)))
7978imbi2d 340 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = (𝑚 + 1) → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑛) ⊆ (𝑔𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗))))
80 ssid 4005 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔𝑗) ⊆ (𝑔𝑗)
81802a1i 12 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ ℤ → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑗) ⊆ (𝑔𝑗)))
82 eluznn 12961 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ)
83 fvoveq1 7455 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
84 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
8583, 84sseq12d 4016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚)))
8685rspccva 3620 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
8782, 86sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ (𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ𝑗))) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
8887anassrs 467 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
89 sstr2 3989 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚) → ((𝑔𝑚) ⊆ (𝑔𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗)))
9088, 89syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝑔𝑚) ⊆ (𝑔𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗)))
9190expcom 413 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (ℤ𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → ((𝑔𝑚) ⊆ (𝑔𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗))))
9291a2d 29 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ𝑗) → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑚) ⊆ (𝑔𝑗)) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗))))
9373, 76, 79, 76, 81, 92uzind4 12949 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑚) ⊆ (𝑔𝑗)))
9493com12 32 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑚 ∈ (ℤ𝑗) → (𝑔𝑚) ⊆ (𝑔𝑗)))
9594ralrimiv 3144 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → ∀𝑚 ∈ (ℤ𝑗)(𝑔𝑚) ⊆ (𝑔𝑗))
9669, 70, 95syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) → ∀𝑚 ∈ (ℤ𝑗)(𝑔𝑚) ⊆ (𝑔𝑗))
97 fveq2 6905 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
9897, 74eleq12d 2834 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 → ((𝑓𝑛) ∈ (𝑔𝑛) ↔ (𝑓𝑚) ∈ (𝑔𝑚)))
99 simplr 768 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))
10099ad2antlr 727 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) ∧ 𝑚 ∈ (ℤ𝑗)) → ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))
10170, 82sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ)
10298, 100, 101rspcdva 3622 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑓𝑚) ∈ (𝑔𝑚))
103102ralrimiva 3145 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑚))
104 r19.26 3110 . . . . . . . . . . . . . . . . . 18 (∀𝑚 ∈ (ℤ𝑗)((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) ↔ (∀𝑚 ∈ (ℤ𝑗)(𝑔𝑚) ⊆ (𝑔𝑗) ∧ ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑚)))
10596, 103, 104sylanbrc 583 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) → ∀𝑚 ∈ (ℤ𝑗)((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)))
106 ssel2 3977 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) → (𝑓𝑚) ∈ (𝑔𝑗))
107106ralimi 3082 . . . . . . . . . . . . . . . . 17 (∀𝑚 ∈ (ℤ𝑗)((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑗))
108105, 107syl 17 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑗))
109 ssel 3976 . . . . . . . . . . . . . . . . 17 ((𝑔𝑗) ⊆ 𝑦 → ((𝑓𝑚) ∈ (𝑔𝑗) → (𝑓𝑚) ∈ 𝑦))
110109ralimdv 3168 . . . . . . . . . . . . . . . 16 ((𝑔𝑗) ⊆ 𝑦 → (∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑗) → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
111108, 110syl5com 31 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ))) → ((𝑔𝑗) ⊆ 𝑦 → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
112111anassrs 467 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) ∧ (𝑦𝐽𝑗 ∈ ℕ)) → ((𝑔𝑗) ⊆ 𝑦 → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
113112anassrs 467 . . . . . . . . . . . . 13 (((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) ∧ 𝑦𝐽) ∧ 𝑗 ∈ ℕ) → ((𝑔𝑗) ⊆ 𝑦 → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
114113reximdva 3167 . . . . . . . . . . . 12 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) ∧ 𝑦𝐽) → (∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
11565, 114syld 47 . . . . . . . . . . 11 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) ∧ 𝑦𝐽) → (𝑃𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
116115ralrimiva 3145 . . . . . . . . . 10 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → ∀𝑦𝐽 (𝑃𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
11736ad2antrr 726 . . . . . . . . . . . 12 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 𝐽 ∈ Top)
1183toptopon 22924 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
119117, 118sylib 218 . . . . . . . . . . 11 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 𝐽 ∈ (TopOn‘𝑋))
120 nnuz 12922 . . . . . . . . . . 11 ℕ = (ℤ‘1)
121 1zzd 12650 . . . . . . . . . . 11 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 1 ∈ ℤ)
122 simprl 770 . . . . . . . . . . . 12 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 𝑓:ℕ⟶𝑆)
12339ad2antrr 726 . . . . . . . . . . . 12 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 𝑆𝑋)
124122, 123fssd 6752 . . . . . . . . . . 11 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 𝑓:ℕ⟶𝑋)
125 eqidd 2737 . . . . . . . . . . 11 ((((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) ∧ 𝑚 ∈ ℕ) → (𝑓𝑚) = (𝑓𝑚))
126119, 120, 121, 124, 125lmbrf 23269 . . . . . . . . . 10 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → (𝑓(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑦𝐽 (𝑃𝑦 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))))
12754, 116, 126mpbir2and 713 . . . . . . . . 9 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ (𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))) → 𝑓(⇝𝑡𝐽)𝑃)
128127expr 456 . . . . . . . 8 (((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) ∧ 𝑓:ℕ⟶𝑆) → (∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛) → 𝑓(⇝𝑡𝐽)𝑃))
129128imdistanda 571 . . . . . . 7 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) → (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
13053, 129syland 603 . . . . . 6 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ((𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) → (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
131130eximdv 1916 . . . . 5 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → (∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) → ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
13250, 131mpd 15 . . . 4 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑔:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) ∧ ∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥))) → ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃))
1338, 132exlimddv 1934 . . 3 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃))
134133ex 412 . 2 ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
1352ad2antrr 726 . . . . . 6 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝐽 ∈ Top)
136135, 118sylib 218 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
137 1zzd 12650 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 1 ∈ ℤ)
138 simprr 772 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓(⇝𝑡𝐽)𝑃)
139 simprl 770 . . . . . 6 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓:ℕ⟶𝑆)
140139ffvelcdmda 7103 . . . . 5 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝑆)
141 simplr 768 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑆𝑋)
142120, 136, 137, 138, 140, 141lmcls 23311 . . . 4 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑃 ∈ ((cls‘𝐽)‘𝑆))
143142ex 412 . . 3 ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → ((𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
144143exlimdv 1932 . 2 ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → (∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
145134, 144impbid 212 1 ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950  c0 4332   cuni 4906   class class class wbr 5142   I cid 5576  wf 6556  cfv 6560  (class class class)co 7432  1c1 11157   + caddc 11159  cn 12267  cz 12615  cuz 12879  Topctop 22900  TopOnctopon 22917  clsccl 23027  𝑡clm 23235  1stωc1stc 23446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cc 10476  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-top 22901  df-topon 22918  df-cld 23028  df-ntr 23029  df-cls 23030  df-lm 23238  df-1stc 23448
This theorem is referenced by:  1stccnp  23471  hausmapdom  23509  1stckgen  23563  metelcls  25340
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