Theorem 2ndcsep 23488

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 23475	. 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2		vex 3448	. . . . . . . . 9 ⊢ 𝑏 ∈ V
3		difss 4080	. . . . . . . . 9 ⊢ (𝑏 ∖ {∅}) ⊆ 𝑏
4		ssdomg 8966	. . . . . . . . 9 ⊢ (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
5	2, 3, 4	mp2 9	. . . . . . . 8 ⊢ (𝑏 ∖ {∅}) ≼ 𝑏
6		simpr 487	. . . . . . . 8 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7		domtr 8973	. . . . . . . 8 ⊢ (((𝑏 ∖ {∅}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
8	5, 6, 7	sylancr 595	. . . . . . 7 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9		eldifsn 4736	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
10		n0 4296	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑦)
11		elunii 4860	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ ∪ 𝑏)
12		simpl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ 𝑦)
13	11, 12	jca 518	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
14	13	expcom 416	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑏 → (𝑧 ∈ 𝑦 → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
15	14	eximdv 1927	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑏 → (∃𝑧 𝑧 ∈ 𝑦 → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
16	15	imp 409	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
17		df-rex 3077	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
18	16, 17	sylibr 236	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
19	10, 18	sylan2b 602	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
20	9, 19	sylbi 219	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
21	20	rgen 3068	. . . . . . 7 ⊢ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦
22		vuniex 7707	. . . . . . . 8 ⊢ ∪ 𝑏 ∈ V
23		eleq1 2840	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑦) → (𝑧 ∈ 𝑦 ↔ (𝑓‘𝑦) ∈ 𝑦))
24	22, 23	axcc4dom 10384	. . . . . . 7 ⊢ (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
25	8, 21, 24	sylancl 594	. . . . . 6 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
26		frn 6684	. . . . . . . . 9 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ran 𝑓 ⊆ ∪ 𝑏)
27	26	ad2antrl 736	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ⊆ ∪ 𝑏)
28		vex 3448	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
29	28	rnex 7876	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
30	29	elpw 4549	. . . . . . . 8 ⊢ (ran 𝑓 ∈ 𝒫 ∪ 𝑏 ↔ ran 𝑓 ⊆ ∪ 𝑏)
31	27, 30	sylibr 236	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 ∪ 𝑏)
32		omelon 9587	. . . . . . . . . . 11 ⊢ ω ∈ On
33	6	adantr 483	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34		ondomen 9979	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
35	32, 33, 34	sylancr 595	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36		ssnum 9981	. . . . . . . . . 10 ⊢ ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
37	35, 3, 36	sylancl 594	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38		ffn 6676	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → 𝑓 Fn (𝑏 ∖ {∅}))
39	38	ad2antrl 736	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40		dffn4 6769	. . . . . . . . . 10 ⊢ (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
41	39, 40	sylib 220	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42		fodomnum 9999	. . . . . . . . 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 483	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45		domtr 8973	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
46	43, 44, 45	syl2anc 592	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47		tgcl 22998	. . . . . . . . . 10 ⊢ (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
48	47	ad2antrr 734	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49		unitg 22996	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ V → ∪ (topGen‘𝑏) = ∪ 𝑏)
50	49	elv 3449	. . . . . . . . . . 11 ⊢ ∪ (topGen‘𝑏) = ∪ 𝑏
51	50	eqcomi 2761	. . . . . . . . . 10 ⊢ ∪ 𝑏 = ∪ (topGen‘𝑏)
52	51	clsss3 23088	. . . . . . . . 9 ⊢ (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 ⊆ ∪ 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
53	48, 27, 52	syl2anc 592	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
54		ne0i 4284	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑦 ≠ ∅)
55	54	anim2i 625	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
56	55, 9	sylibr 236	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57		fnfvelrn 7046	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
58	38, 57	sylan 588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
59		inelcm 4409	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑦) ∈ 𝑦 ∧ (𝑓‘𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
60	59	expcom 416	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 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 434	. . . . . . . . . . . 12 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → (𝑦 ∈ 𝑏 → (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
66	65	ralimdv2 3161	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
67	66	imp 409	. . . . . . . . . 10 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
68	67	ad2antlr 735	. . . . . . . . 9 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69		eqidd 2753	. . . . . . . . . 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 782	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑏 ∈ TopBases)
72	27	adantr 483	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ran 𝑓 ⊆ ∪ 𝑏)
73		simpr 487	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑥 ∈ ∪ 𝑏)
74	69, 70, 71, 72, 73	elcls3 23112	. . . . . . . . 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 259	. . . . . . . 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 3948	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)
77		breq1 5093	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78		fveqeq2 6861	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏))
79	77, 78	anbi12d 640	. . . . . . . 8 ⊢ (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)))
80	79	rspcev 3572	. . . . . . 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 845	. . . . . 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 1945	. . . . 5 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
83		unieq 4866	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ (topGen‘𝑏) = ∪ 𝐽)
84		2ndcsep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
85	83, 51, 84	3eqtr4g 2812	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ 𝑏 = 𝑋)
86	85	pweqd 4562	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → 𝒫 ∪ 𝑏 = 𝒫 𝑋)
87		fveq2 6852	. . . . . . . . 9 ⊢ ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
88	87	fveq1d 6854	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
89	88, 85	eqeq12d 2768	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
90	89	anbi2d 638	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
91	86, 90	rexeqbidv 3327	. . . . 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 247	. . . 4 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
93	92	impr 457	. . 3 ⊢ ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
94	93	rexlimiva 3145	. 2 ⊢ (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
95	1, 94	sylbi 219	1 ⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132   ≠ wne 2947  ∀wral 3066  ∃wrex 3076  Vcvv 3444   ∖ cdif 3892   ∩ cin 3894   ⊆ wss 3895  ∅c0 4276  𝒫 cpw 4545  {csn 4572  ∪ cuni 4855   class class class wbr 5090  dom cdm 5636  ran crn 5637  Oncon0 6331   Fn wfn 6501  ⟶wf 6502  –onto→wfo 6504  ‘cfv 6506  ωcom 7831   ≼ cdom 8910  cardccrd 9879  topGenctg 17438  Topctop 22922  TopBasesctb 22974  clsccl 23047  2^ndωc2ndc 23467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cc 10378

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-iin 4942  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-map 8794  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-card 9883  df-acn 9886  df-topgen 17444  df-top 22923  df-bases 22975  df-cld 23048  df-ntr 23049  df-cls 23050  df-2ndc 23469

This theorem is referenced by:  met2ndc  24552