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Theorem 1stcelcls 21466
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 9449. 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 807 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ 1st𝜔)
2 1stctop 21448 . . . . . . 7 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
3 1stcelcls.1 . . . . . . . 8 𝑋 = 𝐽
43clsss3 21065 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
52, 4sylan 489 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
65sselda 3744 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃𝑋)
731stcfb 21450 . . . . 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 696 . . . 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 simpr1 1234 . . . . . . . . . . 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))) → 𝑔:ℕ⟶𝐽)
109ffvelrnda 6522 . . . . . . . . . 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) → (𝑔𝑛) ∈ 𝐽)
113elcls2 21080 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))))
122, 11sylan 489 . . . . . . . . . . . 12 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))))
1312simplbda 655 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))
1413ad2antrr 764 . . . . . . . . . 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) → ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))
15 simpr2 1236 . . . . . . . . . . . 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)))
16 simpl 474 . . . . . . . . . . . . 13 ((𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → 𝑃 ∈ (𝑔𝑘))
1716ralimi 3090 . . . . . . . . . . . 12 (∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → ∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔𝑘))
1815, 17syl 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))
19 fveq2 6352 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑔𝑘) = (𝑔𝑛))
2019eleq2d 2825 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑃 ∈ (𝑔𝑘) ↔ 𝑃 ∈ (𝑔𝑛)))
2120rspccva 3448 . . . . . . . . . . 11 ((∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔𝑘) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ (𝑔𝑛))
2218, 21sylan 489 . . . . . . . . . 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))
23 eleq2 2828 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → (𝑃𝑦𝑃 ∈ (𝑔𝑛)))
24 ineq1 3950 . . . . . . . . . . . . 13 (𝑦 = (𝑔𝑛) → (𝑦𝑆) = ((𝑔𝑛) ∩ 𝑆))
2524neeq1d 2991 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → ((𝑦𝑆) ≠ ∅ ↔ ((𝑔𝑛) ∩ 𝑆) ≠ ∅))
2623, 25imbi12d 333 . . . . . . . . . . 11 (𝑦 = (𝑔𝑛) → ((𝑃𝑦 → (𝑦𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑔𝑛) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅)))
2726rspcv 3445 . . . . . . . . . 10 ((𝑔𝑛) ∈ 𝐽 → (∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅) → (𝑃 ∈ (𝑔𝑛) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅)))
2810, 14, 22, 27syl3c 66 . . . . . . . . 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 3939 . . . . . . . . . . . 12 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥 ∈ (𝑔𝑛) ∧ 𝑥𝑆))
30 ancom 465 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑔𝑛) ∧ 𝑥𝑆) ↔ (𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3129, 30bitri 264 . . . . . . . . . . 11 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3231exbii 1923 . . . . . . . . . 10 (∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ ∃𝑥(𝑥𝑆𝑥 ∈ (𝑔𝑛)))
33 n0 4074 . . . . . . . . . 10 (((𝑔𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆))
34 df-rex 3056 . . . . . . . . . 10 (∃𝑥𝑆 𝑥 ∈ (𝑔𝑛) ↔ ∃𝑥(𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3532, 33, 343bitr4i 292 . . . . . . . . 9 (((𝑔𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥𝑆 𝑥 ∈ (𝑔𝑛))
3628, 35sylib 208 . . . . . . . 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))
372ad2antrr 764 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
383topopn 20913 . . . . . . . . . . . . 13 (𝐽 ∈ Top → 𝑋𝐽)
3937, 38syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋𝐽)
40 simplr 809 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
4139, 40ssexd 4957 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
42 fvi 6417 . . . . . . . . . . 11 (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
4341, 42syl 17 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ( I ‘𝑆) = 𝑆)
4443ad2antrr 764 . . . . . . . . 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) → ( I ‘𝑆) = 𝑆)
4544rexeqdv 3284 . . . . . . . 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) → (∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔𝑛) ↔ ∃𝑥𝑆 𝑥 ∈ (𝑔𝑛)))
4636, 45mpbird 247 . . . . . . 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))
4746ralrimiva 3104 . . . . . 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))
48 fvex 6362 . . . . . . 7 ( I ‘𝑆) ∈ V
49 nnenom 12973 . . . . . . 7 ℕ ≈ ω
50 eleq1 2827 . . . . . . 7 (𝑥 = (𝑓𝑛) → (𝑥 ∈ (𝑔𝑛) ↔ (𝑓𝑛) ∈ (𝑔𝑛)))
5148, 49, 50axcc4 9453 . . . . . 6 (∀𝑛 ∈ ℕ ∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔𝑛) → ∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)))
5247, 51syl 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)))
5343feq3d 6193 . . . . . . . . 9 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) ↔ 𝑓:ℕ⟶𝑆))
5453biimpd 219 . . . . . . . 8 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) → 𝑓:ℕ⟶𝑆))
5554adantr 472 . . . . . . 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))
566ad2antrr 764 . . . . . . . . . 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)
57 simplr3 1265 . . . . . . . . . . . . 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))
58 eleq2 2828 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
59 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → (𝑔𝑘) = (𝑔𝑗))
6059sseq1d 3773 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → ((𝑔𝑘) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑥))
6160cbvrexv 3311 . . . . . . . . . . . . . . . 16 (∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑥)
62 sseq2 3768 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((𝑔𝑗) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑦))
6362rexbidv 3190 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6461, 63syl5bb 272 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6558, 64imbi12d 333 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥) ↔ (𝑃𝑦 → ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦)))
6665rspccva 3448 . . . . . . . . . . . . 13 ((∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥) ∧ 𝑦𝐽) → (𝑃𝑦 → ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6757, 66sylan 489 . . . . . . . . . . . 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))
68 simpr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘))
6968ralimi 3090 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘))
7015, 69syl 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))
7170adantr 472 . . . . . . . . . . . . . . . . . . 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))
72 simprrr 824 . . . . . . . . . . . . . . . . . . 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)
73 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑗 → (𝑔𝑛) = (𝑔𝑗))
7473sseq1d 3773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑗 → ((𝑔𝑛) ⊆ (𝑔𝑗) ↔ (𝑔𝑗) ⊆ (𝑔𝑗)))
7574imbi2d 329 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑛) ⊆ (𝑔𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑗) ⊆ (𝑔𝑗))))
76 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
7776sseq1d 3773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑔𝑛) ⊆ (𝑔𝑗) ↔ (𝑔𝑚) ⊆ (𝑔𝑗)))
7877imbi2d 329 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑛) ⊆ (𝑔𝑗)) ↔ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑚) ⊆ (𝑔𝑗))))
79 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 + 1) → (𝑔𝑛) = (𝑔‘(𝑚 + 1)))
8079sseq1d 3773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = (𝑚 + 1) → ((𝑔𝑛) ⊆ (𝑔𝑗) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗)))
8180imbi2d 329 . . . . . . . . . . . . . . . . . . . . . 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))))
82 ssid 3765 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔𝑗) ⊆ (𝑔𝑗)
83822a1i 12 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ ℤ → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑗) ⊆ (𝑔𝑗)))
84 eluznn 11951 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ)
85 oveq1 6820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
8685fveq2d 6356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
87 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
8886, 87sseq12d 3775 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚)))
8988rspccva 3448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
9084, 89sylan2 492 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ (𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ𝑗))) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
9190anassrs 683 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
92 sstr2 3751 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚) → ((𝑔𝑚) ⊆ (𝑔𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗)))
9391, 92syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ𝑗)) → ((𝑔𝑚) ⊆ (𝑔𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗)))
9493expcom 450 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (ℤ𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → ((𝑔𝑚) ⊆ (𝑔𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑗))))
9594a2d 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))))
9675, 78, 81, 78, 83, 95uzind4 11939 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑚) ⊆ (𝑔𝑗)))
9796com12 32 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑚 ∈ (ℤ𝑗) → (𝑔𝑚) ⊆ (𝑔𝑗)))
9897ralrimiv 3103 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → ∀𝑚 ∈ (ℤ𝑗)(𝑔𝑚) ⊆ (𝑔𝑗))
9971, 72, 98syl2anc 696 . . . . . . . . . . . . . . . . . 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))
10072, 84sylan 489 . . . . . . . . . . . . . . . . . . . 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)
101 simplr 809 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))
102101ad2antlr 765 . . . . . . . . . . . . . . . . . . . 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))
103 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
104103, 76eleq12d 2833 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → ((𝑓𝑛) ∈ (𝑔𝑛) ↔ (𝑓𝑚) ∈ (𝑔𝑚)))
105104rspcv 3445 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛) → (𝑓𝑚) ∈ (𝑔𝑚)))
106100, 102, 105sylc 65 . . . . . . . . . . . . . . . . . . 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))
107106ralrimiva 3104 . . . . . . . . . . . . . . . . . 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))
108 r19.26 3202 . . . . . . . . . . . . . . . . . 18 (∀𝑚 ∈ (ℤ𝑗)((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) ↔ (∀𝑚 ∈ (ℤ𝑗)(𝑔𝑚) ⊆ (𝑔𝑗) ∧ ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑚)))
10999, 107, 108sylanbrc 701 . . . . . . . . . . . . . . . . 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)))
110 ssel2 3739 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) → (𝑓𝑚) ∈ (𝑔𝑗))
111110ralimi 3090 . . . . . . . . . . . . . . . . 17 (∀𝑚 ∈ (ℤ𝑗)((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑗))
112109, 111syl 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))
113 ssel 3738 . . . . . . . . . . . . . . . . 17 ((𝑔𝑗) ⊆ 𝑦 → ((𝑓𝑚) ∈ (𝑔𝑗) → (𝑓𝑚) ∈ 𝑦))
114113ralimdv 3101 . . . . . . . . . . . . . . . 16 ((𝑔𝑗) ⊆ 𝑦 → (∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑗) → ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ 𝑦))
115112, 114syl5com 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))
116115anassrs 683 . . . . . . . . . . . . . 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))
117116anassrs 683 . . . . . . . . . . . . 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))
118117reximdva 3155 . . . . . . . . . . . 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))
11967, 118syld 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))
120119ralrimiva 3104 . . . . . . . . . 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))
12137ad2antrr 764 . . . . . . . . . . . 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)
1223toptopon 20924 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
123121, 122sylib 208 . . . . . . . . . . 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))
124 nnuz 11916 . . . . . . . . . . 11 ℕ = (ℤ‘1)
125 1zzd 11600 . . . . . . . . . . 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)
126 simprl 811 . . . . . . . . . . . 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)
12740ad2antrr 764 . . . . . . . . . . . 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)
128126, 127fssd 6218 . . . . . . . . . . 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)
129 eqidd 2761 . . . . . . . . . . 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))
130123, 124, 125, 128, 129lmbrf 21266 . . . . . . . . . 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))))
13156, 120, 130mpbir2and 995 . . . . . . . . 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)
132131expr 644 . . . . . . . 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))
133132imdistanda 731 . . . . . . 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)))
13455, 133syland 499 . . . . . 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)))
135134eximdv 1995 . . . . 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)))
13652, 135mpd 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))
1378, 136exlimddv 2012 . . 3 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃))
138137ex 449 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
1392ad2antrr 764 . . . . . 6 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝐽 ∈ Top)
140139, 122sylib 208 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
141 1zzd 11600 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 1 ∈ ℤ)
142 simprr 813 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓(⇝𝑡𝐽)𝑃)
143 simprl 811 . . . . . 6 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓:ℕ⟶𝑆)
144143ffvelrnda 6522 . . . . 5 ((((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝑆)
145 simplr 809 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑆𝑋)
146124, 140, 141, 142, 144, 145lmcls 21308 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑃 ∈ ((cls‘𝐽)‘𝑆))
147146ex 449 . . 3 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → ((𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
148147exlimdv 2010 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → (∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
149138, 148impbid 202 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 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cin 3714  wss 3715  c0 4058   cuni 4588   class class class wbr 4804   I cid 5173  wf 6045  cfv 6049  (class class class)co 6813  1c1 10129   + caddc 10131  cn 11212  cz 11569  cuz 11879  Topctop 20900  TopOnctopon 20917  clsccl 21024  𝑡clm 21232  1st𝜔c1stc 21442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cc 9449  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-top 20901  df-topon 20918  df-cld 21025  df-ntr 21026  df-cls 21027  df-lm 21235  df-1stc 21444
This theorem is referenced by:  1stccnp  21467  hausmapdom  21505  1stckgen  21559  metelcls  23303
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