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Theorem 1stcelcls 23485
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 10473. 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 767 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ 1stω)
2 1stctop 23467 . . . . . . 7 (𝐽 ∈ 1stω → 𝐽 ∈ Top)
3 1stcelcls.1 . . . . . . . 8 𝑋 = 𝐽
43clsss3 23083 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
52, 4sylan 580 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
65sselda 3995 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃𝑋)
731stcfb 23469 . . . . 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 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))) → ∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)))
10 simpl 482 . . . . . . . . . . . . 13 ((𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → 𝑃 ∈ (𝑔𝑘))
1110ralimi 3081 . . . . . . . . . . . 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 6907 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑔𝑘) = (𝑔𝑛))
1413eleq2d 2825 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑃 ∈ (𝑔𝑘) ↔ 𝑃 ∈ (𝑔𝑛)))
1514rspccva 3621 . . . . . . . . . . 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 2828 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → (𝑃𝑦𝑃 ∈ (𝑔𝑛)))
18 ineq1 4221 . . . . . . . . . . . . 13 (𝑦 = (𝑔𝑛) → (𝑦𝑆) = ((𝑔𝑛) ∩ 𝑆))
1918neeq1d 2998 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → ((𝑦𝑆) ≠ ∅ ↔ ((𝑔𝑛) ∩ 𝑆) ≠ ∅))
2017, 19imbi12d 344 . . . . . . . . . . 11 (𝑦 = (𝑔𝑛) → ((𝑃𝑦 → (𝑦𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑔𝑛) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅)))
213elcls2 23098 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . 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 7104 . . . . . . . . . . 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 3623 . . . . . . . . . 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 3979 . . . . . . . . . . . 12 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥 ∈ (𝑔𝑛) ∧ 𝑥𝑆))
3029biancomi 462 . . . . . . . . . . 11 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3130exbii 1845 . . . . . . . . . 10 (∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ ∃𝑥(𝑥𝑆𝑥 ∈ (𝑔𝑛)))
32 n0 4359 . . . . . . . . . 10 (((𝑔𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆))
33 df-rex 3069 . . . . . . . . . 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 22928 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝑋𝐽)
3836, 37syl 17 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋𝐽)
39 simplr 769 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
4038, 39ssexd 5330 . . . . . . . . . 10 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
41 fvi 6985 . . . . . . . . . 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 3329 . . . . . . 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 3144 . . . . . 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 6920 . . . . . . 7 ( I ‘𝑆) ∈ V
47 nnenom 14018 . . . . . . 7 ℕ ≈ ω
48 eleq1 2827 . . . . . . 7 (𝑥 = (𝑓𝑛) → (𝑥 ∈ (𝑔𝑛) ↔ (𝑓𝑛) ∈ (𝑔𝑛)))
4946, 47, 48axcc4 10477 . . . . . 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 6724 . . . . . . . . 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 1216 . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
57 fveq2 6907 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → (𝑔𝑘) = (𝑔𝑗))
5857sseq1d 4027 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → ((𝑔𝑘) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑥))
5958cbvrexvw 3236 . . . . . . . . . . . . . . . 16 (∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑥)
60 sseq2 4022 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((𝑔𝑗) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑦))
6160rexbidv 3177 . . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . 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 3081 . . . . . . . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑗 → (𝑔𝑛) = (𝑔𝑗))
7271sseq1d 4027 . . . . . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
7574sseq1d 4027 . . . . . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 + 1) → (𝑔𝑛) = (𝑔‘(𝑚 + 1)))
7877sseq1d 4027 . . . . . . . . . . . . . . . . . . . . . . 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 4018 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔𝑗) ⊆ (𝑔𝑗)
81802a1i 12 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ ℤ → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑗) ⊆ (𝑔𝑗)))
82 eluznn 12958 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ)
83 fvoveq1 7454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
84 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
8583, 84sseq12d 4029 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚)))
8685rspccva 3621 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4002 . . . . . . . . . . . . . . . . . . . . . . . . 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 12946 . . . . . . . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
9897, 74eleq12d 2833 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 → ((𝑓𝑛) ∈ (𝑔𝑛) ↔ (𝑓𝑚) ∈ (𝑔𝑚)))
99 simplr 769 . . . . . . . . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) → (𝑓𝑚) ∈ (𝑔𝑗))
107106ralimi 3081 . . . . . . . . . . . . . . . . 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 3989 . . . . . . . . . . . . . . . . 17 ((𝑔𝑗) ⊆ 𝑦 → ((𝑓𝑚) ∈ (𝑔𝑗) → (𝑓𝑚) ∈ 𝑦))
110109ralimdv 3167 . . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . 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 3144 . . . . . . . . . 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 22939 . . . . . . . . . . . 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 12919 . . . . . . . . . . 11 ℕ = (ℤ‘1)
121 1zzd 12646 . . . . . . . . . . 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 771 . . . . . . . . . . . 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 6754 . . . . . . . . . . 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 2736 . . . . . . . . . . 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 23284 . . . . . . . . . 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 1915 . . . . 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 1933 . . 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 12646 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 1 ∈ ℤ)
138 simprr 773 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓(⇝𝑡𝐽)𝑃)
139 simprl 771 . . . . . 6 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓:ℕ⟶𝑆)
140139ffvelcdmda 7104 . . . . 5 ((((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝑆)
141 simplr 769 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑆𝑋)
142120, 136, 137, 138, 140, 141lmcls 23326 . . . 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 1931 . 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 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  c0 4339   cuni 4912   class class class wbr 5148   I cid 5582  wf 6559  cfv 6563  (class class class)co 7431  1c1 11154   + caddc 11156  cn 12264  cz 12611  cuz 12876  Topctop 22915  TopOnctopon 22932  clsccl 23042  𝑡clm 23250  1stωc1stc 23461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-top 22916  df-topon 22933  df-cld 23043  df-ntr 23044  df-cls 23045  df-lm 23253  df-1stc 23463
This theorem is referenced by:  1stccnp  23486  hausmapdom  23524  1stckgen  23578  metelcls  25353
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