1stcelcls - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 1stcelcls		Structured version Visualization version GIF version

Theorem 1stcelcls 22965

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 10430. 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 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)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.

Step	Hyp	Ref	Expression
1		simpll 766	. . . . 5 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ 1^stω)
2		1stctop 22947	. . . . . . 7 ⊢ (𝐽 ∈ 1^stω → 𝐽 ∈ Top)
3		1stcelcls.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
4	3	clsss3 22563	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
5	2, 4	sylan 581	. . . . . 6 ⊢ ((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
6	5	sselda 3983	. . . . 5 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ 𝑋)
7	3	1stcfb 22949	. . . . 5 ⊢ ((𝐽 ∈ 1^stω ∧ 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 585	. . . 4 ⊢ (((𝐽 ∈ 1^stω ∧ 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 1196	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1^stω ∧ 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 484	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → 𝑃 ∈ (𝑔‘𝑘))
11	10	ralimi 3084	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → ∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔‘𝑘))
12	9, 11	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1^stω ∧ 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 6892	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛))
14	13	eleq2d 2820	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑃 ∈ (𝑔‘𝑘) ↔ 𝑃 ∈ (𝑔‘𝑛)))
15	14	rspccva 3612	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ ℕ 𝑃 ∈ (𝑔‘𝑘) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ (𝑔‘𝑛))
16	12, 15	sylan 581	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ 1^stω ∧ 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 2823	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑔‘𝑛) → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ (𝑔‘𝑛)))
18		ineq1 4206	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑔‘𝑛) → (𝑦 ∩ 𝑆) = ((𝑔‘𝑛) ∩ 𝑆))
19	18	neeq1d 3001	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑔‘𝑛) → ((𝑦 ∩ 𝑆) ≠ ∅ ↔ ((𝑔‘𝑛) ∩ 𝑆) ≠ ∅))
20	17, 19	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑔‘𝑛) → ((𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑔‘𝑛) → ((𝑔‘𝑛) ∩ 𝑆) ≠ ∅)))
21	3	elcls2 22578	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅))))
22	2, 21	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅))))
23	22	simplbda 501	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 → (𝑦 ∩ 𝑆) ≠ ∅))
24	23	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ 1^stω ∧ 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 1195	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1^stω ∧ 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 7087	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ 1^stω ∧ 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 3614	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ 1^stω ∧ 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 ∈ 1^stω ∧ 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 3965	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆) ↔ (𝑥 ∈ (𝑔‘𝑛) ∧ 𝑥 ∈ 𝑆))
30	29	biancomi 464	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑔‘𝑛)))
31	30	exbii 1851	. . . . . . . . . 10 ⊢ (∃𝑥 𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑔‘𝑛)))
32		n0 4347	. . . . . . . . . 10 ⊢ (((𝑔‘𝑛) ∩ 𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑔‘𝑛) ∩ 𝑆))
33		df-rex 3072	. . . . . . . . . 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 217	. . . . . . . 8 ⊢ (((((𝐽 ∈ 1^stω ∧ 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 725	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
37	3	topopn 22408	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
38	36, 37	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 ∈ 𝐽)
39		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
40	38, 39	ssexd 5325	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
41		fvi 6968	. . . . . . . . . . 11 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ( I ‘𝑆) = 𝑆)
43	42	ad2antrr 725	. . . . . . . . 9 ⊢ (((((𝐽 ∈ 1^stω ∧ 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	43	rexeqdv 3327	. . . . . . . 8 ⊢ (((((𝐽 ∈ 1^stω ∧ 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) ↔ ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑔‘𝑛)))
45	35, 44	mpbird 257	. . . . . . 7 ⊢ (((((𝐽 ∈ 1^stω ∧ 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	45	ralrimiva 3147	. . . . . 6 ⊢ ((((𝐽 ∈ 1^stω ∧ 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))
47		fvex 6905	. . . . . . 7 ⊢ ( I ‘𝑆) ∈ V
48		nnenom 13945	. . . . . . 7 ⊢ ℕ ≈ ω
49		eleq1 2822	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ∈ (𝑔‘𝑛) ↔ (𝑓‘𝑛) ∈ (𝑔‘𝑛)))
50	47, 48, 49	axcc4 10434	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ ∃𝑥 ∈ ( I ‘𝑆)𝑥 ∈ (𝑔‘𝑛) → ∃𝑓(𝑓:ℕ⟶( I ‘𝑆) ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)))
51	46, 50	syl 17	. . . . 5 ⊢ ((((𝐽 ∈ 1^stω ∧ 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)))
52	42	feq3d 6705	. . . . . . . . 9 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) ↔ 𝑓:ℕ⟶𝑆))
53	52	biimpd 228	. . . . . . . 8 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑓:ℕ⟶( I ‘𝑆) → 𝑓:ℕ⟶𝑆))
54	53	adantr 482	. . . . . . 7 ⊢ ((((𝐽 ∈ 1^stω ∧ 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))
55	6	ad2antrr 725	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)
56		simplr3 1218	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
57		eleq2 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
58		fveq2 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑗 → (𝑔‘𝑘) = (𝑔‘𝑗))
59	58	sseq1d 4014	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑗 → ((𝑔‘𝑘) ⊆ 𝑥 ↔ (𝑔‘𝑗) ⊆ 𝑥))
60	59	cbvrexvw 3236	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑥)
61		sseq2 4009	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑗) ⊆ 𝑥 ↔ (𝑔‘𝑗) ⊆ 𝑦))
62	61	rexbidv 3179	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦))
63	60, 62	bitrid 283	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥 ↔ ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦))
64	57, 63	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥) ↔ (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦)))
65	64	rspccva 3612	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑥) ∧ 𝑦 ∈ 𝐽) → (𝑃 ∈ 𝑦 → ∃𝑗 ∈ ℕ (𝑔‘𝑗) ⊆ 𝑦))
66	56, 65	sylan 581	. . . . . . . . . . . 12 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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))
67		simpr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘))
68	67	ralimi 3084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑘 ∈ ℕ (𝑃 ∈ (𝑔‘𝑘) ∧ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘)) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘))
69	9, 68	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ 1^stω ∧ 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))
70	69	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
71		simprrr 781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)
72		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑗 → (𝑔‘𝑛) = (𝑔‘𝑗))
73	72	sseq1d 4014	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑗 → ((𝑔‘𝑛) ⊆ (𝑔‘𝑗) ↔ (𝑔‘𝑗) ⊆ (𝑔‘𝑗)))
74	73	imbi2d 341	. . . . . . . . . . . . . . . . . . . . . 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))))
75		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
76	75	sseq1d 4014	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → ((𝑔‘𝑛) ⊆ (𝑔‘𝑗) ↔ (𝑔‘𝑚) ⊆ (𝑔‘𝑗)))
77	76	imbi2d 341	. . . . . . . . . . . . . . . . . . . . . 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))))
78		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = (𝑚 + 1) → (𝑔‘𝑛) = (𝑔‘(𝑚 + 1)))
79	78	sseq1d 4014	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = (𝑚 + 1) → ((𝑔‘𝑛) ⊆ (𝑔‘𝑗) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗)))
80	79	imbi2d 341	. . . . . . . . . . . . . . . . . . . . . 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))))
81		ssid 4005	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔‘𝑗) ⊆ (𝑔‘𝑗)
82	81	2a1i 12	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ ℤ → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑗) ⊆ (𝑔‘𝑗)))
83		eluznn 12902	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ ℕ)
84		fvoveq1 7432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
85		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑚 → (𝑔‘𝑘) = (𝑔‘𝑚))
86	84, 85	sseq12d 4016	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ↔ (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚)))
87	86	rspccva 3612	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚))
88	83, 87	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ (𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚))
89	88	anassrs 469	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚))
90		sstr2 3990	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑚) → ((𝑔‘𝑚) ⊆ (𝑔‘𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗)))
91	89, 90	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → ((𝑔‘𝑚) ⊆ (𝑔‘𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗)))
92	91	expcom 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → ((𝑔‘𝑚) ⊆ (𝑔‘𝑗) → (𝑔‘(𝑚 + 1)) ⊆ (𝑔‘𝑗))))
93	92	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))))
94	74, 77, 80, 77, 82, 93	uzind4 12890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) → ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑔‘𝑚) ⊆ (𝑔‘𝑗)))
95	94	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → (𝑚 ∈ (ℤ_≥‘𝑗) → (𝑔‘𝑚) ⊆ (𝑔‘𝑗)))
96	95	ralrimiv 3146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ⊆ (𝑔‘𝑘) ∧ 𝑗 ∈ ℕ) → ∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑔‘𝑚) ⊆ (𝑔‘𝑗))
97	70, 71, 96	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
98		fveq2 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
99	98, 75	eleq12d 2828	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑚 → ((𝑓‘𝑛) ∈ (𝑔‘𝑛) ↔ (𝑓‘𝑚) ∈ (𝑔‘𝑚)))
100		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:ℕ⟶𝑆 ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ (𝑓‘𝑛) ∈ (𝑔‘𝑛))
101	100	ad2antlr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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))
102	71, 83	sylan 581	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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)
103	99, 101, 102	rspcdva 3614	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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	103	ralrimiva 3147	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
105		r19.26 3112	. . . . . . . . . . . . . . . . . 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)))
106	97, 104, 105	sylanbrc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)))
107		ssel2 3978	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑚) ⊆ (𝑔‘𝑗) ∧ (𝑓‘𝑚) ∈ (𝑔‘𝑚)) → (𝑓‘𝑚) ∈ (𝑔‘𝑗))
108	107	ralimi 3084	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)((𝑔‘𝑚) ⊆ (𝑔‘𝑗) ∧ (𝑓‘𝑚) ∈ (𝑔‘𝑚)) → ∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑓‘𝑚) ∈ (𝑔‘𝑗))
109	106, 108	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
110		ssel 3976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔‘𝑗) ⊆ 𝑦 → ((𝑓‘𝑚) ∈ (𝑔‘𝑗) → (𝑓‘𝑚) ∈ 𝑦))
111	110	ralimdv 3170	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔‘𝑗) ⊆ 𝑦 → (∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑓‘𝑚) ∈ (𝑔‘𝑗) → ∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑓‘𝑚) ∈ 𝑦))
112	109, 111	syl5com 31	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ 1^stω ∧ 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 469	. . . . . . . . . . . . . 14 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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	anassrs 469	. . . . . . . . . . . . 13 ⊢ (((((((𝐽 ∈ 1^stω ∧ 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))
115	114	reximdva 3169	. . . . . . . . . . . 12 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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))
116	66, 115	syld 47	. . . . . . . . . . 11 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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	116	ralrimiva 3147	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
118	36	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)
119	3	toptopon 22419	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
120	118, 119	sylib 217	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
121		nnuz 12865	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
122		1zzd 12593	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)
123		simprl 770	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)
124	39	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)
125	123, 124	fssd 6736	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ 1^stω ∧ 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)
126		eqidd 2734	. . . . . . . . . . 11 ⊢ ((((((𝐽 ∈ 1^stω ∧ 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))
127	120, 121, 122, 125, 126	lmbrf 22764	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))))
128	55, 117, 127	mpbir2and 712	. . . . . . . . 9 ⊢ (((((𝐽 ∈ 1^stω ∧ 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	expr 458	. . . . . . . 8 ⊢ (((((𝐽 ∈ 1^stω ∧ 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))
130	129	imdistanda 573	. . . . . . 7 ⊢ ((((𝐽 ∈ 1^stω ∧ 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)))
131	54, 130	syland 604	. . . . . 6 ⊢ ((((𝐽 ∈ 1^stω ∧ 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)))
132	131	eximdv 1921	. . . . 5 ⊢ ((((𝐽 ∈ 1^stω ∧ 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)))
133	51, 132	mpd 15	. . . 4 ⊢ ((((𝐽 ∈ 1^stω ∧ 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))
134	8, 133	exlimddv 1939	. . 3 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃))
135	134	ex 414	. 2 ⊢ ((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)))
136	2	ad2antrr 725	. . . . . 6 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) → 𝐽 ∈ Top)
137	136, 119	sylib 217	. . . . 5 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
138		1zzd 12593	. . . . 5 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) → 1 ∈ ℤ)
139		simprr 772	. . . . 5 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) → 𝑓(⇝_𝑡‘𝐽)𝑃)
140		simprl 770	. . . . . 6 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) → 𝑓:ℕ⟶𝑆)
141	140	ffvelcdmda 7087	. . . . 5 ⊢ ((((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝑆)
142		simplr 768	. . . . 5 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) → 𝑆 ⊆ 𝑋)
143	121, 137, 138, 139, 141, 142	lmcls 22806	. . . 4 ⊢ (((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) ∧ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)) → 𝑃 ∈ ((cls‘𝐽)‘𝑆))
144	143	ex 414	. . 3 ⊢ ((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) → ((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
145	144	exlimdv 1937	. 2 ⊢ ((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) → (∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃) → 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
146	135, 145	impbid 211	1 ⊢ ((𝐽 ∈ 1^stω ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝_𝑡‘𝐽)𝑃)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 ∪ cuni 4909 class class class wbr 5149 I cid 5574 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1c1 11111 + caddc 11113 ℕcn 12212 ℤcz 12558 ℤ_≥cuz 12822 Topctop 22395 TopOnctopon 22412 clsccl 22522 ⇝_𝑡clm 22730 1^stωc1stc 22941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-top 22396 df-topon 22413 df-cld 22523 df-ntr 22524 df-cls 22525 df-lm 22733 df-1stc 22943

This theorem is referenced by: 1stccnp 22966 hausmapdom 23004 1stckgen 23058 metelcls 24822

W3C validator