Theorem 2ndcsep 23595

Description: A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)

Hypothesis

Ref	Expression
2ndcsep.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
2ndcsep	⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem 2ndcsep

Dummy variables 𝑓 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		is2ndc 23582	. 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2		vex 3457	. . . . . . . . 9 ⊢ 𝑏 ∈ V
3		difss 4089	. . . . . . . . 9 ⊢ (𝑏 ∖ {∅}) ⊆ 𝑏
4		ssdomg 8996	. . . . . . . . 9 ⊢ (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
5	2, 3, 4	mp2 9	. . . . . . . 8 ⊢ (𝑏 ∖ {∅}) ≼ 𝑏
6		simpr 489	. . . . . . . 8 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7		domtr 9003	. . . . . . . 8 ⊢ (((𝑏 ∖ {∅}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
8	5, 6, 7	sylancr 598	. . . . . . 7 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9		eldifsn 4752	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
10		n0 4306	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑦)
11		elunii 4876	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ ∪ 𝑏)
12		simpl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ 𝑦)
13	11, 12	jca 520	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
14	13	expcom 418	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑏 → (𝑧 ∈ 𝑦 → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
15	14	eximdv 1945	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑏 → (∃𝑧 𝑧 ∈ 𝑦 → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
16	15	imp 411	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
17		df-rex 3088	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
18	16, 17	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
19	10, 18	sylan2b 605	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
20	9, 19	sylbi 220	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
21	20	rgen 3079	. . . . . . 7 ⊢ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦
22		vuniex 7737	. . . . . . . 8 ⊢ ∪ 𝑏 ∈ V
23		eleq1 2849	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑦) → (𝑧 ∈ 𝑦 ↔ (𝑓‘𝑦) ∈ 𝑦))
24	22, 23	axcc4dom 10424	. . . . . . 7 ⊢ (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
25	8, 21, 24	sylancl 597	. . . . . 6 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
26		frn 6713	. . . . . . . . 9 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ran 𝑓 ⊆ ∪ 𝑏)
27	26	ad2antrl 740	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ⊆ ∪ 𝑏)
28		vex 3457	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
29	28	rnex 7906	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
30	29	elpw 4565	. . . . . . . 8 ⊢ (ran 𝑓 ∈ 𝒫 ∪ 𝑏 ↔ ran 𝑓 ⊆ ∪ 𝑏)
31	27, 30	sylibr 237	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 ∪ 𝑏)
32		omelon 9614	. . . . . . . . . . 11 ⊢ ω ∈ On
33	6	adantr 485	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34		ondomen 10020	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
35	32, 33, 34	sylancr 598	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36		ssnum 10022	. . . . . . . . . 10 ⊢ ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
37	35, 3, 36	sylancl 597	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38		ffn 6705	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → 𝑓 Fn (𝑏 ∖ {∅}))
39	38	ad2antrl 740	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40		dffn4 6798	. . . . . . . . . 10 ⊢ (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
41	39, 40	sylib 221	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42		fodomnum 10040	. . . . . . . . 9 ⊢ ((𝑏 ∖ {∅}) ∈ dom card → (𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓 → ran 𝑓 ≼ (𝑏 ∖ {∅})))
43	37, 41, 42	sylc 66	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ (𝑏 ∖ {∅}))
44	8	adantr 485	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45		domtr 9003	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
46	43, 44, 45	syl2anc 595	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47		tgcl 23105	. . . . . . . . . 10 ⊢ (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
48	47	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49		unitg 23103	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ V → ∪ (topGen‘𝑏) = ∪ 𝑏)
50	49	elv 3458	. . . . . . . . . . 11 ⊢ ∪ (topGen‘𝑏) = ∪ 𝑏
51	50	eqcomi 2770	. . . . . . . . . 10 ⊢ ∪ 𝑏 = ∪ (topGen‘𝑏)
52	51	clsss3 23195	. . . . . . . . 9 ⊢ (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 ⊆ ∪ 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
53	48, 27, 52	syl2anc 595	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
54		ne0i 4293	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑦 ≠ ∅)
55	54	anim2i 628	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
56	55, 9	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57		fnfvelrn 7075	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
58	38, 57	sylan 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
59		inelcm 4424	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑦) ∈ 𝑦 ∧ (𝑓‘𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
60	59	expcom 418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑦) ∈ ran 𝑓 → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
61	58, 60	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
62	61	ex 417	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → (𝑦 ∈ (𝑏 ∖ {∅}) → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
63	62	a2d 30	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → (𝑦 ∈ (𝑏 ∖ {∅}) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
64	56, 63	syl7 75	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
65	64	exp4a 436	. . . . . . . . . . . 12 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → (𝑦 ∈ 𝑏 → (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
66	65	ralimdv2 3172	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
67	66	imp 411	. . . . . . . . . 10 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
68	67	ad2antlr 739	. . . . . . . . 9 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69		eqidd 2762	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → (topGen‘𝑏) = (topGen‘𝑏))
70	51	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ∪ 𝑏 = ∪ (topGen‘𝑏))
71		simplll 786	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑏 ∈ TopBases)
72	27	adantr 485	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ran 𝑓 ⊆ ∪ 𝑏)
73		simpr 489	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑥 ∈ ∪ 𝑏)
74	69, 70, 71, 72, 73	elcls3 23219	. . . . . . . . 9 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → (𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓) ↔ ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
75	68, 74	mpbird 260	. . . . . . . 8 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓))
76	53, 75	eqelssd 3957	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)
77		breq1 5111	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78		fveqeq2 6890	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏))
79	77, 78	anbi12d 643	. . . . . . . 8 ⊢ (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)))
80	79	rspcev 3580	. . . . . . 7 ⊢ ((ran 𝑓 ∈ 𝒫 ∪ 𝑏 ∧ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
81	31, 46, 76, 80	syl12anc 849	. . . . . 6 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
82	25, 81	exlimddv 1963	. . . . 5 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
83		unieq 4882	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ (topGen‘𝑏) = ∪ 𝐽)
84		2ndcsep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
85	83, 51, 84	3eqtr4g 2821	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ 𝑏 = 𝑋)
86	85	pweqd 4578	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → 𝒫 ∪ 𝑏 = 𝒫 𝑋)
87		fveq2 6881	. . . . . . . . 9 ⊢ ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
88	87	fveq1d 6883	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
89	88, 85	eqeq12d 2777	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
90	89	anbi2d 641	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
91	86, 90	rexeqbidv 3337	. . . . 5 ⊢ ((topGen‘𝑏) = 𝐽 → (∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
92	82, 91	syl5ibcom 248	. . . 4 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
93	92	impr 459	. . 3 ⊢ ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
94	93	rexlimiva 3156	. 2 ⊢ (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
95	1, 94	sylbi 220	1 ⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3453   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  {csn 4588  ∪ cuni 4871   class class class wbr 5108  dom cdm 5661  ran crn 5662  Oncon0 6360   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  ωcom 7861   ≼ cdom 8940  cardccrd 9920  topGenctg 17489  Topctop 23029  TopBasesctb 23081  clsccl 23154  2^ndωc2ndc 23574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cc 10418

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9924  df-acn 9927  df-topgen 17495  df-top 23030  df-bases 23082  df-cld 23155  df-ntr 23156  df-cls 23157  df-2ndc 23576

This theorem is referenced by:  met2ndc  24659