2ndcomap - Metamath Proof Explorer

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Theorem 2ndcomap 23487

Description: A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.)

**Hypotheses**
Ref	Expression
2ndcomap.2	⊢ 𝑌 = ∪ 𝐾
2ndcomap.3	⊢ (𝜑 → 𝐽 ∈ 2^ndω)
2ndcomap.5	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
2ndcomap.6	⊢ (𝜑 → ran 𝐹 = 𝑌)
2ndcomap.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)

**Assertion**
Ref	Expression
2ndcomap	⊢ (𝜑 → 𝐾 ∈ 2^ndω)

Distinct variable groups: 𝑥,𝐹 𝑥,𝐽 𝜑,𝑥 𝑥,𝐾 Allowed substitution hint: 𝑌(𝑥)

Proof of Theorem 2ndcomap Dummy variables 𝑘 𝑚 𝑡 𝑤 𝑧 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2ndcomap.5	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
2		cntop2 23270	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
4	3	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ Top)
5		simplll 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → 𝜑)
6		bastg 22994	. . . . . . . . . 10 ⊢ (𝑏 ∈ TopBases → 𝑏 ⊆ (topGen‘𝑏))
7	6	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ (topGen‘𝑏))
8		simprr 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘𝑏) = 𝐽)
9	7, 8	sseqtrd 4049	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ 𝐽)
10	9	sselda 4008	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ 𝐽)
11		2ndcomap.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
12	5, 10, 11	syl2anc 583	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → (𝐹 “ 𝑥) ∈ 𝐾)
13	12	fmpttd 7149	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏⟶𝐾)
14	13	frnd 6755	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ⊆ 𝐾)
15		elunii 4936	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑘 ∧ 𝑘 ∈ 𝐾) → 𝑧 ∈ ∪ 𝐾)
16		2ndcomap.2	. . . . . . . . . . 11 ⊢ 𝑌 = ∪ 𝐾
17	15, 16	eleqtrrdi 2855	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑘 ∧ 𝑘 ∈ 𝐾) → 𝑧 ∈ 𝑌)
18	17	ancoms 458	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) → 𝑧 ∈ 𝑌)
19	18	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝑧 ∈ 𝑌)
20		2ndcomap.6	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝑌)
21	20	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ran 𝐹 = 𝑌)
22	19, 21	eleqtrrd 2847	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝑧 ∈ ran 𝐹)
23		eqid 2740	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23, 16	cnf 23275	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
25	1, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌)
26	25	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝐹:∪ 𝐽⟶𝑌)
27		ffn 6747	. . . . . . . 8 ⊢ (𝐹:∪ 𝐽⟶𝑌 → 𝐹 Fn ∪ 𝐽)
28		fvelrnb 6982	. . . . . . . 8 ⊢ (𝐹 Fn ∪ 𝐽 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑡 ∈ ∪ 𝐽(𝐹‘𝑡) = 𝑧))
29	26, 27, 28	3syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑡 ∈ ∪ 𝐽(𝐹‘𝑡) = 𝑧))
30	22, 29	mpbid 232	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ∃𝑡 ∈ ∪ 𝐽(𝐹‘𝑡) = 𝑧)
31	1	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝐹 ∈ (𝐽 Cn 𝐾))
32		simprll 778	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑘 ∈ 𝐾)
33		cnima 23294	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑘 ∈ 𝐾) → (^◡𝐹 “ 𝑘) ∈ 𝐽)
34	31, 32, 33	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (^◡𝐹 “ 𝑘) ∈ 𝐽)
35	8	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (topGen‘𝑏) = 𝐽)
36	34, 35	eleqtrrd 2847	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (^◡𝐹 “ 𝑘) ∈ (topGen‘𝑏))
37		simprrl 780	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑡 ∈ ∪ 𝐽)
38		simprrr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝐹‘𝑡) = 𝑧)
39		simprlr 779	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑧 ∈ 𝑘)
40	38, 39	eqeltrd 2844	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝐹‘𝑡) ∈ 𝑘)
41	26	ffnd 6748	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝐹 Fn ∪ 𝐽)
42	41	adantrr 716	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝐹 Fn ∪ 𝐽)
43		elpreima 7091	. . . . . . . . . . 11 ⊢ (𝐹 Fn ∪ 𝐽 → (𝑡 ∈ (^◡𝐹 “ 𝑘) ↔ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) ∈ 𝑘)))
44	42, 43	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝑡 ∈ (^◡𝐹 “ 𝑘) ↔ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) ∈ 𝑘)))
45	37, 40, 44	mpbir2and 712	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑡 ∈ (^◡𝐹 “ 𝑘))
46		tg2 22993	. . . . . . . . 9 ⊢ (((^◡𝐹 “ 𝑘) ∈ (topGen‘𝑏) ∧ 𝑡 ∈ (^◡𝐹 “ 𝑘)) → ∃𝑚 ∈ 𝑏 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))
47	36, 45, 46	syl2anc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → ∃𝑚 ∈ 𝑏 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))
48		simprl 770	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → 𝑚 ∈ 𝑏)
49		eqid 2740	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑚) = (𝐹 “ 𝑚)
50		imaeq2 6085	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑚 → (𝐹 “ 𝑥) = (𝐹 “ 𝑚))
51	50	rspceeqv 3658	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑏 ∧ (𝐹 “ 𝑚) = (𝐹 “ 𝑚)) → ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥))
52	48, 49, 51	sylancl 585	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥))
53	42	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → 𝐹 Fn ∪ 𝐽)
54		fnfun 6679	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn ∪ 𝐽 → Fun 𝐹)
55	53, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → Fun 𝐹)
56		simprrr 781	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → 𝑚 ⊆ (^◡𝐹 “ 𝑘))
57		funimass2 6661	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)) → (𝐹 “ 𝑚) ⊆ 𝑘)
58	55, 56, 57	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → (𝐹 “ 𝑚) ⊆ 𝑘)
59		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑘 ∈ V
60		ssexg 5341	. . . . . . . . . . . 12 ⊢ (((𝐹 “ 𝑚) ⊆ 𝑘 ∧ 𝑘 ∈ V) → (𝐹 “ 𝑚) ∈ V)
61	58, 59, 60	sylancl 585	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → (𝐹 “ 𝑚) ∈ V)
62		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) = (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))
63	62	elrnmpt 5981	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑚) ∈ V → ((𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥)))
64	61, 63	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → ((𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥)))
65	52, 64	mpbird 257	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → (𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)))
66	38	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → (𝐹‘𝑡) = 𝑧)
67		simprrl 780	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → 𝑡 ∈ 𝑚)
68		cnvimass 6111	. . . . . . . . . . . . 13 ⊢ (^◡𝐹 “ 𝑘) ⊆ dom 𝐹
69	56, 68	sstrdi 4021	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → 𝑚 ⊆ dom 𝐹)
70		funfvima2 7268	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑚 ⊆ dom 𝐹) → (𝑡 ∈ 𝑚 → (𝐹‘𝑡) ∈ (𝐹 “ 𝑚)))
71	55, 69, 70	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → (𝑡 ∈ 𝑚 → (𝐹‘𝑡) ∈ (𝐹 “ 𝑚)))
72	67, 71	mpd 15	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → (𝐹‘𝑡) ∈ (𝐹 “ 𝑚))
73	66, 72	eqeltrrd 2845	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → 𝑧 ∈ (𝐹 “ 𝑚))
74		eleq2 2833	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 “ 𝑚) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝐹 “ 𝑚)))
75		sseq1 4034	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 “ 𝑚) → (𝑤 ⊆ 𝑘 ↔ (𝐹 “ 𝑚) ⊆ 𝑘))
76	74, 75	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹 “ 𝑚) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘) ↔ (𝑧 ∈ (𝐹 “ 𝑚) ∧ (𝐹 “ 𝑚) ⊆ 𝑘)))
77	76	rspcev 3635	. . . . . . . . 9 ⊢ (((𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∧ (𝑧 ∈ (𝐹 “ 𝑚) ∧ (𝐹 “ 𝑚) ⊆ 𝑘)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘))
78	65, 73, 58, 77	syl12anc 836	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘))
79	47, 78	rexlimddv 3167	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘))
80	79	anassrs 467	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘))
81	30, 80	rexlimddv 3167	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘))
82	81	ralrimivva 3208	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∀𝑘 ∈ 𝐾 ∀𝑧 ∈ 𝑘 ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘))
83		basgen2 23017	. . . 4 ⊢ ((𝐾 ∈ Top ∧ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ⊆ 𝐾 ∧ ∀𝑘 ∈ 𝐾 ∀𝑧 ∈ 𝑘 ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) = 𝐾)
84	4, 14, 82, 83	syl3anc 1371	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) = 𝐾)
85	84, 4	eqeltrd 2844	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ Top)
86		tgclb 22998	. . . . 5 ⊢ (ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ Top)
87	85, 86	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases)
88		omelon 9715	. . . . . . 7 ⊢ ω ∈ On
89		simprl 770	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ≼ ω)
90		ondomen 10106	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
91	88, 89, 90	sylancr 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ∈ dom card)
92	13	ffnd 6748	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) Fn 𝑏)
93		dffn4 6840	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) Fn 𝑏 ↔ (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)))
94	92, 93	sylib 218	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)))
95		fodomnum 10126	. . . . . 6 ⊢ (𝑏 ∈ dom card → ((𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ 𝑏))
96	91, 94, 95	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ 𝑏)
97		domtr 9067	. . . . 5 ⊢ ((ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ 𝑏 ∧ 𝑏 ≼ ω) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω)
98	96, 89, 97	syl2anc 583	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω)
99		2ndci 23477	. . . 4 ⊢ ((ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases ∧ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ 2^ndω)
100	87, 98, 99	syl2anc 583	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ 2^ndω)
101	84, 100	eqeltrrd 2845	. 2 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ 2^ndω)
102		2ndcomap.3	. . 3 ⊢ (𝜑 → 𝐽 ∈ 2^ndω)
103		is2ndc 23475	. . 3 ⊢ (𝐽 ∈ 2^ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
104	102, 103	sylib 218	. 2 ⊢ (𝜑 → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
105	101, 104	r19.29a 3168	1 ⊢ (𝜑 → 𝐾 ∈ 2^ndω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ⊆ wss 3976 ∪ cuni 4931 class class class wbr 5166 ↦ cmpt 5249 ^◡ccnv 5699 dom cdm 5700 ran crn 5701 “ cima 5703 Oncon0 6395 Fun wfun 6567 Fn wfn 6568 ⟶wf 6569 –onto→wfo 6571 ‘cfv 6573 (class class class)co 7448 ωcom 7903 ≼ cdom 9001 cardccrd 10004 topGenctg 17497 Topctop 22920 TopBasesctb 22973 Cn ccn 23253 2^ndωc2ndc 23467

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-card 10008 df-acn 10011 df-topgen 17503 df-top 22921 df-topon 22938 df-bases 22974 df-cn 23256 df-2ndc 23469

This theorem is referenced by: (None)

W3C validator