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Theorem kqnrmlem1 23869

Description: A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)

**Hypothesis**
Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

**Assertion**
Ref	Expression
kqnrmlem1	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)

Distinct variable groups: 𝑥,𝑦,𝐽 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem kqnrmlem1 Dummy variables 𝑚 𝑤 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqtopon 23853	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3	2	adantr 485	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4		topontop 23039	. . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
5	3, 4	syl 18	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Top)
6		simplr 780	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐽 ∈ Nrm)
7	1	kqid 23854	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
8	7	ad2antrr 738	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
9		simprl 782	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑧 ∈ (KQ‘𝐽))
10		cnima 23391	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑧 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑧) ∈ 𝐽)
11	8, 9, 10	syl2anc 595	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (^◡𝐹 “ 𝑧) ∈ 𝐽)
12		simprr 784	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))
13	12	elin1d 4165	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
14		cnclima 23394	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑤 ∈ (Clsd‘(KQ‘𝐽))) → (^◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽))
15	8, 13, 14	syl2anc 595	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (^◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽))
16	12	elin2d 4166	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
17		elpwi 4574	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧)
18		imass2 6105	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (^◡𝐹 “ 𝑤) ⊆ (^◡𝐹 “ 𝑧))
19	16, 17, 18	3syl 19	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (^◡𝐹 “ 𝑤) ⊆ (^◡𝐹 “ 𝑧))
20		nrmsep3 23481	. . . . 5 ⊢ ((𝐽 ∈ Nrm ∧ ((^◡𝐹 “ 𝑧) ∈ 𝐽 ∧ (^◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽) ∧ (^◡𝐹 “ 𝑤) ⊆ (^◡𝐹 “ 𝑧))) → ∃𝑢 ∈ 𝐽 ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))
21	6, 11, 15, 19, 20	syl13anc 1397	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑢 ∈ 𝐽 ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))
22		simplll 786	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
23		simprl 782	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑢 ∈ 𝐽)
24	1	kqopn 23860	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐽) → (𝐹 “ 𝑢) ∈ (KQ‘𝐽))
25	22, 23, 24	syl2anc 595	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → (𝐹 “ 𝑢) ∈ (KQ‘𝐽))
26		simprrl 792	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → (^◡𝐹 “ 𝑤) ⊆ 𝑢)
27	1	kqffn 23851	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
28		fnfun 6636	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
29	22, 27, 28	3syl 19	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → Fun 𝐹)
30	13	adantr 485	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
31		eqid 2769	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
32	31	cldss 23155	. . . . . . . . 9 ⊢ (𝑤 ∈ (Clsd‘(KQ‘𝐽)) → 𝑤 ⊆ ∪ (KQ‘𝐽))
33	30, 32	syl 18	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑤 ⊆ ∪ (KQ‘𝐽))
34		toponuni 23040	. . . . . . . . 9 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
35	22, 2, 34	3syl 19	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ran 𝐹 = ∪ (KQ‘𝐽))
36	33, 35	sseqtrrd 3982	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑤 ⊆ ran 𝐹)
37		funimass1 6619	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ ran 𝐹) → ((^◡𝐹 “ 𝑤) ⊆ 𝑢 → 𝑤 ⊆ (𝐹 “ 𝑢)))
38	29, 36, 37	syl2anc 595	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((^◡𝐹 “ 𝑤) ⊆ 𝑢 → 𝑤 ⊆ (𝐹 “ 𝑢)))
39	26, 38	mpd 16	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑤 ⊆ (𝐹 “ 𝑢))
40		topontop 23039	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
41	22, 40	syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝐽 ∈ Top)
42		elssuni 4908	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽)
43	42	ad2antrl 740	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑢 ⊆ ∪ 𝐽)
44		eqid 2769	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
45	44	clscld 23173	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46	41, 43, 45	syl2anc 595	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
47	1	kqcld 23861	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)))
48	22, 46, 47	syl2anc 595	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)))
49	44	sscls 23182	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
50	41, 43, 49	syl2anc 595	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
51		imass2 6105	. . . . . . . 8 ⊢ (𝑢 ⊆ ((cls‘𝐽)‘𝑢) → (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
52	50, 51	syl 18	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
53	31	clsss2 23198	. . . . . . 7 ⊢ (((𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
54	48, 52, 53	syl2anc 595	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
55		simprrr 793	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧))
56	44	clsss3 23185	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)
57	41, 43, 56	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)
58		fndm 6639	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
59	22, 27, 58	3syl 19	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → dom 𝐹 = 𝑋)
60		toponuni 23040	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
61	22, 60	syl 18	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → 𝑋 = ∪ 𝐽)
62	59, 61	eqtrd 2804	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → dom 𝐹 = ∪ 𝐽)
63	57, 62	sseqtrrd 3982	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹)
64		funimass3 7050	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))
65	29, 63, 64	syl2anc 595	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))
66	55, 65	mpbird 260	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧)
67	54, 66	sstrd 3955	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)
68		sseq2 3971	. . . . . . 7 ⊢ (𝑚 = (𝐹 “ 𝑢) → (𝑤 ⊆ 𝑚 ↔ 𝑤 ⊆ (𝐹 “ 𝑢)))
69		fveq2 6882	. . . . . . . 8 ⊢ (𝑚 = (𝐹 “ 𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)))
70	69	sseq1d 3976	. . . . . . 7 ⊢ (𝑚 = (𝐹 “ 𝑢) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧))
71	68, 70	anbi12d 643	. . . . . 6 ⊢ (𝑚 = (𝐹 “ 𝑢) → ((𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹 “ 𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)))
72	71	rspcev 3590	. . . . 5 ⊢ (((𝐹 “ 𝑢) ∈ (KQ‘𝐽) ∧ (𝑤 ⊆ (𝐹 “ 𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
73	25, 39, 67, 72	syl12anc 849	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((^◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
74	21, 73	rexlimddv 3178	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
75	74	ralrimivva 3214	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → ∀𝑧 ∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
76		isnrm 23461	. 2 ⊢ ((KQ‘𝐽) ∈ Nrm ↔ ((KQ‘𝐽) ∈ Top ∧ ∀𝑧 ∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)))
77	5, 75, 76	sylanbrc 594	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 {crab 3423 ∩ cin 3912 ⊆ wss 3913 𝒫 cpw 4567 ∪ cuni 4876 ↦ cmpt 5196 ^◡ccnv 5661 dom cdm 5662 ran crn 5663 “ cima 5665 Fun wfun 6531 Fn wfn 6532 ‘cfv 6537 (class class class)co 7411 Topctop 23019 TopOnctopon 23036 Clsdccld 23142 clsccl 23144 Cn ccn 23350 Nrmcnrm 23436 KQckq 23819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8826 df-qtop 17561 df-top 23020 df-topon 23037 df-cld 23145 df-cls 23147 df-cn 23353 df-nrm 23443 df-kq 23820

This theorem is referenced by: kqnrm 23878

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