Theorem 2ndcsep 23412

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 23399	. 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2		vex 3431	. . . . . . . . 9 ⊢ 𝑏 ∈ V
3		difss 4068	. . . . . . . . 9 ⊢ (𝑏 ∖ {∅}) ⊆ 𝑏
4		ssdomg 8936	. . . . . . . . 9 ⊢ (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
5	2, 3, 4	mp2 9	. . . . . . . 8 ⊢ (𝑏 ∖ {∅}) ≼ 𝑏
6		simpr 484	. . . . . . . 8 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7		domtr 8943	. . . . . . . 8 ⊢ (((𝑏 ∖ {∅}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
8	5, 6, 7	sylancr 588	. . . . . . 7 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9		eldifsn 4721	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
10		n0 4283	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑦)
11		elunii 4845	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ ∪ 𝑏)
12		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ 𝑦)
13	11, 12	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
14	13	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑏 → (𝑧 ∈ 𝑦 → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
15	14	eximdv 1919	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑏 → (∃𝑧 𝑧 ∈ 𝑦 → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
16	15	imp 406	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
17		df-rex 3060	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
18	16, 17	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
19	10, 18	sylan2b 595	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
20	9, 19	sylbi 217	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
21	20	rgen 3051	. . . . . . 7 ⊢ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦
22		vuniex 7682	. . . . . . . 8 ⊢ ∪ 𝑏 ∈ V
23		eleq1 2823	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑦) → (𝑧 ∈ 𝑦 ↔ (𝑓‘𝑦) ∈ 𝑦))
24	22, 23	axcc4dom 10352	. . . . . . 7 ⊢ (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
25	8, 21, 24	sylancl 587	. . . . . 6 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
26		frn 6664	. . . . . . . . 9 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ran 𝑓 ⊆ ∪ 𝑏)
27	26	ad2antrl 729	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ⊆ ∪ 𝑏)
28		vex 3431	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
29	28	rnex 7850	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
30	29	elpw 4535	. . . . . . . 8 ⊢ (ran 𝑓 ∈ 𝒫 ∪ 𝑏 ↔ ran 𝑓 ⊆ ∪ 𝑏)
31	27, 30	sylibr 234	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 ∪ 𝑏)
32		omelon 9556	. . . . . . . . . . 11 ⊢ ω ∈ On
33	6	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34		ondomen 9948	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
35	32, 33, 34	sylancr 588	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36		ssnum 9950	. . . . . . . . . 10 ⊢ ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
37	35, 3, 36	sylancl 587	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38		ffn 6657	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → 𝑓 Fn (𝑏 ∖ {∅}))
39	38	ad2antrl 729	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40		dffn4 6747	. . . . . . . . . 10 ⊢ (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
41	39, 40	sylib 218	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42		fodomnum 9968	. . . . . . . . 9 ⊢ ((𝑏 ∖ {∅}) ∈ dom card → (𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓 → ran 𝑓 ≼ (𝑏 ∖ {∅})))
43	37, 41, 42	sylc 65	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ (𝑏 ∖ {∅}))
44	8	adantr 480	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45		domtr 8943	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
46	43, 44, 45	syl2anc 585	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47		tgcl 22922	. . . . . . . . . 10 ⊢ (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
48	47	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49		unitg 22920	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ V → ∪ (topGen‘𝑏) = ∪ 𝑏)
50	49	elv 3432	. . . . . . . . . . 11 ⊢ ∪ (topGen‘𝑏) = ∪ 𝑏
51	50	eqcomi 2744	. . . . . . . . . 10 ⊢ ∪ 𝑏 = ∪ (topGen‘𝑏)
52	51	clsss3 23012	. . . . . . . . 9 ⊢ (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 ⊆ ∪ 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
53	48, 27, 52	syl2anc 585	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
54		ne0i 4271	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑦 ≠ ∅)
55	54	anim2i 618	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
56	55, 9	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57		fnfvelrn 7021	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
58	38, 57	sylan 581	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
59		inelcm 4395	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑦) ∈ 𝑦 ∧ (𝑓‘𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
60	59	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑦) ∈ ran 𝑓 → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
61	58, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
62	61	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → (𝑦 ∈ (𝑏 ∖ {∅}) → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
63	62	a2d 29	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → (𝑦 ∈ (𝑏 ∖ {∅}) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
64	56, 63	syl7 74	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
65	64	exp4a 431	. . . . . . . . . . . 12 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → (𝑦 ∈ 𝑏 → (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
66	65	ralimdv2 3144	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
67	66	imp 406	. . . . . . . . . 10 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
68	67	ad2antlr 728	. . . . . . . . 9 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69		eqidd 2736	. . . . . . . . . 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 775	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑏 ∈ TopBases)
72	27	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ran 𝑓 ⊆ ∪ 𝑏)
73		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑥 ∈ ∪ 𝑏)
74	69, 70, 71, 72, 73	elcls3 23036	. . . . . . . . 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 257	. . . . . . . 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 3938	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)
77		breq1 5077	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78		fveqeq2 6838	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏))
79	77, 78	anbi12d 633	. . . . . . . 8 ⊢ (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)))
80	79	rspcev 3562	. . . . . . 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 837	. . . . . 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 1937	. . . . 5 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
83		unieq 4851	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ (topGen‘𝑏) = ∪ 𝐽)
84		2ndcsep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
85	83, 51, 84	3eqtr4g 2795	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ 𝑏 = 𝑋)
86	85	pweqd 4548	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → 𝒫 ∪ 𝑏 = 𝒫 𝑋)
87		fveq2 6829	. . . . . . . . 9 ⊢ ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
88	87	fveq1d 6831	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
89	88, 85	eqeq12d 2751	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
90	89	anbi2d 631	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
91	86, 90	rexeqbidv 3310	. . . . 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 245	. . . 4 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
93	92	impr 454	. . 3 ⊢ ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
94	93	rexlimiva 3128	. 2 ⊢ (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
95	1, 94	sylbi 217	1 ⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2930  ∀wral 3049  ∃wrex 3059  Vcvv 3427   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  ∅c0 4263  𝒫 cpw 4531  {csn 4557  ∪ cuni 4840   class class class wbr 5074  dom cdm 5620  ran crn 5621  Oncon0 6312   Fn wfn 6482  ⟶wf 6483  –onto→wfo 6485  ‘cfv 6487  ωcom 7806   ≼ cdom 8880  cardccrd 9848  topGenctg 17389  Topctop 22846  TopBasesctb 22898  clsccl 22971  2^ndωc2ndc 23391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-inf2 9551  ax-cc 10346

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-iin 4926  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8632  df-map 8764  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-card 9852  df-acn 9855  df-topgen 17395  df-top 22847  df-bases 22899  df-cld 22972  df-ntr 22973  df-cls 22974  df-2ndc 23393

This theorem is referenced by:  met2ndc  24476