Theorem 2ndcsep 22518

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 22505	. 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2		vex 3426	. . . . . . . . 9 ⊢ 𝑏 ∈ V
3		difss 4062	. . . . . . . . 9 ⊢ (𝑏 ∖ {∅}) ⊆ 𝑏
4		ssdomg 8741	. . . . . . . . 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 8748	. . . . . . . 8 ⊢ (((𝑏 ∖ {∅}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
8	5, 6, 7	sylancr 586	. . . . . . 7 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9		eldifsn 4717	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
10		n0 4277	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑦)
11		elunii 4841	. . . . . . . . . . . . . . 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 1921	. . . . . . . . . . . 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 3069	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
18	16, 17	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
19	10, 18	sylan2b 593	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
20	9, 19	sylbi 216	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
21	20	rgen 3073	. . . . . . 7 ⊢ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦
22		vuniex 7570	. . . . . . . 8 ⊢ ∪ 𝑏 ∈ V
23		eleq1 2826	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑦) → (𝑧 ∈ 𝑦 ↔ (𝑓‘𝑦) ∈ 𝑦))
24	22, 23	axcc4dom 10128	. . . . . . 7 ⊢ (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
25	8, 21, 24	sylancl 585	. . . . . 6 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
26		frn 6591	. . . . . . . . 9 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ran 𝑓 ⊆ ∪ 𝑏)
27	26	ad2antrl 724	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ⊆ ∪ 𝑏)
28		vex 3426	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
29	28	rnex 7733	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
30	29	elpw 4534	. . . . . . . 8 ⊢ (ran 𝑓 ∈ 𝒫 ∪ 𝑏 ↔ ran 𝑓 ⊆ ∪ 𝑏)
31	27, 30	sylibr 233	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 ∪ 𝑏)
32		omelon 9334	. . . . . . . . . . 11 ⊢ ω ∈ On
33	6	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34		ondomen 9724	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
35	32, 33, 34	sylancr 586	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36		ssnum 9726	. . . . . . . . . 10 ⊢ ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
37	35, 3, 36	sylancl 585	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38		ffn 6584	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → 𝑓 Fn (𝑏 ∖ {∅}))
39	38	ad2antrl 724	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40		dffn4 6678	. . . . . . . . . 10 ⊢ (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
41	39, 40	sylib 217	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42		fodomnum 9744	. . . . . . . . 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 8748	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
46	43, 44, 45	syl2anc 583	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47		tgcl 22027	. . . . . . . . . 10 ⊢ (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
48	47	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49		unitg 22025	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ V → ∪ (topGen‘𝑏) = ∪ 𝑏)
50	49	elv 3428	. . . . . . . . . . 11 ⊢ ∪ (topGen‘𝑏) = ∪ 𝑏
51	50	eqcomi 2747	. . . . . . . . . 10 ⊢ ∪ 𝑏 = ∪ (topGen‘𝑏)
52	51	clsss3 22118	. . . . . . . . 9 ⊢ (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 ⊆ ∪ 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
53	48, 27, 52	syl2anc 583	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
54		ne0i 4265	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑦 ≠ ∅)
55	54	anim2i 616	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
56	55, 9	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57		fnfvelrn 6940	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
58	38, 57	sylan 579	. . . . . . . . . . . . . . . . 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 3101	. . . . . . . . . . 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 723	. . . . . . . . 9 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69		eqidd 2739	. . . . . . . . . 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 771	. . . . . . . . . 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 22142	. . . . . . . . 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 256	. . . . . . . 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 5073	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78		fveqeq2 6765	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏))
79	77, 78	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)))
80	79	rspcev 3552	. . . . . . 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 833	. . . . . 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 1939	. . . . 5 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
83		unieq 4847	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ (topGen‘𝑏) = ∪ 𝐽)
84		2ndcsep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
85	83, 51, 84	3eqtr4g 2804	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ 𝑏 = 𝑋)
86	85	pweqd 4549	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → 𝒫 ∪ 𝑏 = 𝒫 𝑋)
87		fveq2 6756	. . . . . . . . 9 ⊢ ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
88	87	fveq1d 6758	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
89	88, 85	eqeq12d 2754	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
90	89	anbi2d 628	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
91	86, 90	rexeqbidv 3328	. . . . 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 244	. . . 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 3209	. 2 ⊢ (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
95	1, 94	sylbi 216	1 ⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   class class class wbr 5070  dom cdm 5580  ran crn 5581  Oncon0 6251   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  ωcom 7687   ≼ cdom 8689  cardccrd 9624  topGenctg 17065  Topctop 21950  TopBasesctb 22003  clsccl 22077  2^ndωc2ndc 22497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cc 10122

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-acn 9631  df-topgen 17071  df-top 21951  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-2ndc 22499

This theorem is referenced by:  met2ndc  23585