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Theorem 1stcelcls 21074
 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 9140. 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 786 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ 1st𝜔)
2 1stctop 21056 . . . . . . 7 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
3 1stcelcls.1 . . . . . . . 8 𝑋 = 𝐽
43clsss3 20673 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
52, 4sylan 487 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
65sselda 3568 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃𝑋)
731stcfb 21058 . . . . 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 691 . . . 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 1060 . . . . . . . . . . 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 6267 . . . . . . . . . 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 20688 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))))
122, 11sylan 487 . . . . . . . . . . . 12 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))))
1312simplbda 652 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅))
1413ad2antrr 758 . . . . . . . . . 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 1061 . . . . . . . . . . . 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 472 . . . . . . . . . . . . 13 ((𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → 𝑃 ∈ (𝑔𝑘))
1716ralimi 2936 . . . . . . . . . . . 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 6103 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑔𝑘) = (𝑔𝑛))
2019eleq2d 2673 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑃 ∈ (𝑔𝑘) ↔ 𝑃 ∈ (𝑔𝑛)))
2120rspccva 3281 . . . . . . . . . . 11 ((∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔𝑘) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ (𝑔𝑛))
2218, 21sylan 487 . . . . . . . . . 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 2677 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → (𝑃𝑦𝑃 ∈ (𝑔𝑛)))
24 ineq1 3769 . . . . . . . . . . . . 13 (𝑦 = (𝑔𝑛) → (𝑦𝑆) = ((𝑔𝑛) ∩ 𝑆))
2524neeq1d 2841 . . . . . . . . . . . 12 (𝑦 = (𝑔𝑛) → ((𝑦𝑆) ≠ ∅ ↔ ((𝑔𝑛) ∩ 𝑆) ≠ ∅))
2623, 25imbi12d 333 . . . . . . . . . . 11 (𝑦 = (𝑔𝑛) → ((𝑃𝑦 → (𝑦𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑔𝑛) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅)))
2726rspcv 3278 . . . . . . . . . 10 ((𝑔𝑛) ∈ 𝐽 → (∀𝑦𝐽 (𝑃𝑦 → (𝑦𝑆) ≠ ∅) → (𝑃 ∈ (𝑔𝑛) → ((𝑔𝑛) ∩ 𝑆) ≠ ∅)))
2810, 14, 22, 27syl3c 64 . . . . . . . . 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 3758 . . . . . . . . . . . 12 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥 ∈ (𝑔𝑛) ∧ 𝑥𝑆))
30 ancom 465 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑔𝑛) ∧ 𝑥𝑆) ↔ (𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3129, 30bitri 263 . . . . . . . . . . 11 (𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ (𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3231exbii 1764 . . . . . . . . . 10 (∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆) ↔ ∃𝑥(𝑥𝑆𝑥 ∈ (𝑔𝑛)))
33 n0 3890 . . . . . . . . . 10 (((𝑔𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑔𝑛) ∩ 𝑆))
34 df-rex 2902 . . . . . . . . . 10 (∃𝑥𝑆 𝑥 ∈ (𝑔𝑛) ↔ ∃𝑥(𝑥𝑆𝑥 ∈ (𝑔𝑛)))
3532, 33, 343bitr4i 291 . . . . . . . . 9 (((𝑔𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥𝑆 𝑥 ∈ (𝑔𝑛))
3628, 35sylib 207 . . . . . . . 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 758 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
383topopn 20536 . . . . . . . . . . . . 13 (𝐽 ∈ Top → 𝑋𝐽)
3937, 38syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋𝐽)
40 simplr 788 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
4139, 40ssexd 4733 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
42 fvi 6165 . . . . . . . . . . 11 (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
4341, 42syl 17 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ( I ‘𝑆) = 𝑆)
4443ad2antrr 758 . . . . . . . . 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 3122 . . . . . . . 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 246 . . . . . . 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 2949 . . . . . 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 6113 . . . . . . 7 ( I ‘𝑆) ∈ V
49 nnenom 12641 . . . . . . 7 ℕ ≈ ω
50 eleq1 2676 . . . . . . 7 (𝑥 = (𝑓𝑛) → (𝑥 ∈ (𝑔𝑛) ↔ (𝑓𝑛) ∈ (𝑔𝑛)))
5148, 49, 50axcc4 9144 . . . . . 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 5945 . . . . . . . . 9 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) ↔ 𝑓:ℕ⟶𝑆))
5453biimpd 218 . . . . . . . 8 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) → 𝑓:ℕ⟶𝑆))
5554adantr 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))
566ad2antrr 758 . . . . . . . . . 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 1098 . . . . . . . . . . . . 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 2677 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
59 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → (𝑔𝑘) = (𝑔𝑗))
6059sseq1d 3595 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → ((𝑔𝑘) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑥))
6160cbvrexv 3148 . . . . . . . . . . . . . . . 16 (∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑥)
62 sseq2 3590 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((𝑔𝑗) ⊆ 𝑥 ↔ (𝑔𝑗) ⊆ 𝑦))
6362rexbidv 3034 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6461, 63syl5bb 271 . . . . . . . . . . . . . . 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 3281 . . . . . . . . . . . . 13 ((∀𝑥𝐽 (𝑃𝑥 → ∃𝑘 ∈ ℕ (𝑔𝑘) ⊆ 𝑥) ∧ 𝑦𝐽) → (𝑃𝑦 → ∃𝑗 ∈ ℕ (𝑔𝑗) ⊆ 𝑦))
6757, 66sylan 487 . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃 ∈ (𝑔𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘)) → (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘))
6968ralimi 2936 . . . . . . . . . . . . . . . . . . . . 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 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))
72 simprrr 801 . . . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑗 → (𝑔𝑛) = (𝑔𝑗))
7473sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
7776sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 + 1) → (𝑔𝑛) = (𝑔‘(𝑚 + 1)))
8079sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . 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 3587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔𝑗) ⊆ (𝑔𝑗)
83822a1i 12 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ ℤ → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔𝑗) ⊆ (𝑔𝑗)))
84 eluznn 11634 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ𝑗)) → 𝑚 ∈ ℕ)
85 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
8685fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
87 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
8886, 87sseq12d 3597 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚)))
8988rspccva 3281 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
9084, 89sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ (𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ𝑗))) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
9190anassrs 678 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ𝑗)) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔𝑚))
92 sstr2 3575 . . . . . . . . . . . . . . . . . . . . . . . . 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 11622 . . . . . . . . . . . . . . . . . . . . 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 2948 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔𝑘) ∧ 𝑗 ∈ ℕ) → ∀𝑚 ∈ (ℤ𝑗)(𝑔𝑚) ⊆ (𝑔𝑗))
9971, 72, 98syl2anc 691 . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛)) ∧ (𝑦𝐽𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛))
102101ad2antlr 759 . . . . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
104103, 76eleq12d 2682 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → ((𝑓𝑛) ∈ (𝑔𝑛) ↔ (𝑓𝑚) ∈ (𝑔𝑚)))
105104rspcv 3278 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (∀𝑛 ∈ ℕ (𝑓𝑛) ∈ (𝑔𝑛) → (𝑓𝑚) ∈ (𝑔𝑚)))
106100, 102, 105sylc 63 . . . . . . . . . . . . . . . . . . 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 2949 . . . . . . . . . . . . . . . . . 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 3046 . . . . . . . . . . . . . . . . . 18 (∀𝑚 ∈ (ℤ𝑗)((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) ↔ (∀𝑚 ∈ (ℤ𝑗)(𝑔𝑚) ⊆ (𝑔𝑗) ∧ ∀𝑚 ∈ (ℤ𝑗)(𝑓𝑚) ∈ (𝑔𝑚)))
10999, 107, 108sylanbrc 695 . . . . . . . . . . . . . . . . 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 3563 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑚) ⊆ (𝑔𝑗) ∧ (𝑓𝑚) ∈ (𝑔𝑚)) → (𝑓𝑚) ∈ (𝑔𝑗))
111110ralimi 2936 . . . . . . . . . . . . . . . . 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 3562 . . . . . . . . . . . . . . . . 17 ((𝑔𝑗) ⊆ 𝑦 → ((𝑓𝑚) ∈ (𝑔𝑗) → (𝑓𝑚) ∈ 𝑦))
114113ralimdv 2946 . . . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . 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 3000 . . . . . . . . . . . 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 46 . . . . . . . . . . 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 2949 . . . . . . . . . 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 758 . . . . . . . . . . . 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 20548 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
123121, 122sylib 207 . . . . . . . . . . 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 11599 . . . . . . . . . . 11 ℕ = (ℤ‘1)
125 1zzd 11285 . . . . . . . . . . 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 790 . . . . . . . . . . . 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 758 . . . . . . . . . . . 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 5970 . . . . . . . . . . 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 2611 . . . . . . . . . . 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 20874 . . . . . . . . . 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 959 . . . . . . . . 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 641 . . . . . . . 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 725 . . . . . . 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 497 . . . . . 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 1833 . . . . 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 1850 . . 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 758 . . . . . 6 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝐽 ∈ Top)
140139, 122sylib 207 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
141 1zzd 11285 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 1 ∈ ℤ)
142 simprr 792 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓(⇝𝑡𝐽)𝑃)
143 simprl 790 . . . . . 6 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑓:ℕ⟶𝑆)
144143ffvelrnda 6267 . . . . 5 ((((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝑆)
145 simplr 788 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) ∧ (𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)) → 𝑆𝑋)
146124, 140, 141, 142, 144, 145lmcls 20916 . . . 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 1848 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → (∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
149138, 148impbid 201 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372   class class class wbr 4583   I cid 4948  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  ℕcn 10897  ℤcz 11254  ℤ≥cuz 11563  Topctop 20517  TopOnctopon 20518  clsccl 20632  ⇝𝑡clm 20840  1st𝜔c1stc 21050 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-lm 20843  df-1stc 21052 This theorem is referenced by:  1stccnp  21075  hausmapdom  21113  1stckgen  21167  metelcls  22911
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