2ndcomap - Metamath Proof Explorer

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Theorem 2ndcomap 23448

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 23231	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
4	3	ad2antrr 732	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ Top)
5		simplll 780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → 𝜑)
6		bastg 22956	. . . . . . . . . 10 ⊢ (𝑏 ∈ TopBases → 𝑏 ⊆ (topGen‘𝑏))
7	6	ad2antlr 733	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ (topGen‘𝑏))
8		simprr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘𝑏) = 𝐽)
9	7, 8	sseqtrd 3958	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ 𝐽)
10	9	sselda 3922	. . . . . . 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 590	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → (𝐹 “ 𝑥) ∈ 𝐾)
13	12	fmpttd 7063	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏⟶𝐾)
14	13	frnd 6670	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ⊆ 𝐾)
15		elunii 4850	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑘 ∧ 𝑘 ∈ 𝐾) → 𝑧 ∈ ∪ 𝐾)
16		2ndcomap.2	. . . . . . . . . . 11 ⊢ 𝑌 = ∪ 𝐾
17	15, 16	eleqtrrdi 2851	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑘 ∧ 𝑘 ∈ 𝐾) → 𝑧 ∈ 𝑌)
18	17	ancoms 459	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) → 𝑧 ∈ 𝑌)
19	18	adantl 482	. . . . . . . 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 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ran 𝐹 = 𝑌)
22	19, 21	eleqtrrd 2843	. . . . . . 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 23236	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
25	1, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌)
26	25	ad3antrrr 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝐹:∪ 𝐽⟶𝑌)
27		ffn 6662	. . . . . . . 8 ⊢ (𝐹:∪ 𝐽⟶𝑌 → 𝐹 Fn ∪ 𝐽)
28		fvelrnb 6894	. . . . . . . 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 233	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ∃𝑡 ∈ ∪ 𝐽(𝐹‘𝑡) = 𝑧)
31	1	ad3antrrr 736	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝐹 ∈ (𝐽 Cn 𝐾))
32		simprll 784	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑘 ∈ 𝐾)
33		cnima 23255	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑘 ∈ 𝐾) → (^◡𝐹 “ 𝑘) ∈ 𝐽)
34	31, 32, 33	syl2anc 590	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (^◡𝐹 “ 𝑘) ∈ 𝐽)
35	8	adantr 481	. . . . . . . . . 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 2843	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (^◡𝐹 “ 𝑘) ∈ (topGen‘𝑏))
37		simprrl 786	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑡 ∈ ∪ 𝐽)
38		simprrr 787	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝐹‘𝑡) = 𝑧)
39		simprlr 785	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑧 ∈ 𝑘)
40	38, 39	eqeltrd 2840	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝐹‘𝑡) ∈ 𝑘)
41	26	ffnd 6663	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝐹 Fn ∪ 𝐽)
42	41	adantrr 723	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝐹 Fn ∪ 𝐽)
43		elpreima 7006	. . . . . . . . . . 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 719	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑡 ∈ (^◡𝐹 “ 𝑘))
46		tg2 22955	. . . . . . . . 9 ⊢ (((^◡𝐹 “ 𝑘) ∈ (topGen‘𝑏) ∧ 𝑡 ∈ (^◡𝐹 “ 𝑘)) → ∃𝑚 ∈ 𝑏 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)))
47	36, 45, 46	syl2anc 590	. . . . . . . 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 776	. . . . . . . . . . 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 6015	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑚 → (𝐹 “ 𝑥) = (𝐹 “ 𝑚))
51	50	rspceeqv 3590	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑏 ∧ (𝐹 “ 𝑚) = (𝐹 “ 𝑚)) → ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥))
52	48, 49, 51	sylancl 592	. . . . . . . . . 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 481	. . . . . . . . . . . . . 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 6592	. . . . . . . . . . . . . 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 787	. . . . . . . . . . . . 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 6575	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑚 ⊆ (^◡𝐹 “ 𝑘)) → (𝐹 “ 𝑚) ⊆ 𝑘)
58	55, 56, 57	syl2anc 590	. . . . . . . . . . . 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 3436	. . . . . . . . . . . 12 ⊢ 𝑘 ∈ V
60		ssexg 5258	. . . . . . . . . . . 12 ⊢ (((𝐹 “ 𝑚) ⊆ 𝑘 ∧ 𝑘 ∈ V) → (𝐹 “ 𝑚) ∈ V)
61	58, 59, 60	sylancl 592	. . . . . . . . . . 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 5907	. . . . . . . . . . 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 258	. . . . . . . . 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 481	. . . . . . . . . 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 786	. . . . . . . . . . 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 6041	. . . . . . . . . . . . 13 ⊢ (^◡𝐹 “ 𝑘) ⊆ dom 𝐹
69	56, 68	sstrdi 3934	. . . . . . . . . . . 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 7182	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑚 ⊆ dom 𝐹) → (𝑡 ∈ 𝑚 → (𝐹‘𝑡) ∈ (𝐹 “ 𝑚)))
71	55, 69, 70	syl2anc 590	. . . . . . . . . . 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 2841	. . . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 “ 𝑚) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝐹 “ 𝑚)))
75		sseq1 3947	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 “ 𝑚) → (𝑤 ⊆ 𝑘 ↔ (𝐹 “ 𝑚) ⊆ 𝑘))
76	74, 75	anbi12d 638	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹 “ 𝑚) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘) ↔ (𝑧 ∈ (𝐹 “ 𝑚) ∧ (𝐹 “ 𝑚) ⊆ 𝑘)))
77	76	rspcev 3567	. . . . . . . . 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 842	. . . . . . . 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 3147	. . . . . . 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 468	. . . . . 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 3147	. . . . 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 3183	. . . 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 22979	. . . 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 1379	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) = 𝐾)
85	84, 4	eqeltrd 2840	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ Top)
86		tgclb 22960	. . . . 5 ⊢ (ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ Top)
87	85, 86	sylibr 235	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases)
88		omelon 9565	. . . . . . 7 ⊢ ω ∈ On
89		simprl 776	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ≼ ω)
90		ondomen 9957	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
91	88, 89, 90	sylancr 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ∈ dom card)
92	13	ffnd 6663	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) Fn 𝑏)
93		dffn4 6752	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) Fn 𝑏 ↔ (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)))
94	92, 93	sylib 219	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)))
95		fodomnum 9977	. . . . . 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 8951	. . . . 5 ⊢ ((ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ 𝑏 ∧ 𝑏 ≼ ω) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω)
98	96, 89, 97	syl2anc 590	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω)
99		2ndci 23438	. . . 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 590	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ 2^ndω)
101	84, 100	eqeltrrd 2841	. 2 ⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ 2^ndω)
102		2ndcomap.3	. . 3 ⊢ (𝜑 → 𝐽 ∈ 2^ndω)
103		is2ndc 23436	. . 3 ⊢ (𝐽 ∈ 2^ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
104	102, 103	sylib 219	. 2 ⊢ (𝜑 → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
105	101, 104	r19.29a 3148	1 ⊢ (𝜑 → 𝐾 ∈ 2^ndω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3054 ∃wrex 3064 Vcvv 3432 ⊆ wss 3890 ∪ cuni 4845 class class class wbr 5079 ↦ cmpt 5160 ^◡ccnv 5624 dom cdm 5625 ran crn 5626 “ cima 5628 Oncon0 6317 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 –onto→wfo 6490 ‘cfv 6492 (class class class)co 7363 ωcom 7813 ≼ cdom 8888 cardccrd 9857 topGenctg 17398 Topctop 22883 TopBasesctb 22935 Cn ccn 23214 2^ndωc2ndc 23428

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-card 9861 df-acn 9864 df-topgen 17404 df-top 22884 df-topon 22901 df-bases 22936 df-cn 23217 df-2ndc 23430

This theorem is referenced by: (None)

W3C validator