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Theorem kqnrmlem1 23685

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 23669	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3	2	adantr 480	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4		topontop 22855	. . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
5	3, 4	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Top)
6		simplr 768	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐽 ∈ Nrm)
7	1	kqid 23670	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
8	7	ad2antrr 726	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
9		simprl 770	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑧 ∈ (KQ‘𝐽))
10		cnima 23207	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑧 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑧) ∈ 𝐽)
11	8, 9, 10	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (^◡𝐹 “ 𝑧) ∈ 𝐽)
12		simprr 772	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))
13	12	elin1d 4154	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
14		cnclima 23210	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑤 ∈ (Clsd‘(KQ‘𝐽))) → (^◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽))
15	8, 13, 14	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (^◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽))
16	12	elin2d 4155	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
17		elpwi 4559	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧)
18		imass2 6059	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (^◡𝐹 “ 𝑤) ⊆ (^◡𝐹 “ 𝑧))
19	16, 17, 18	3syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (^◡𝐹 “ 𝑤) ⊆ (^◡𝐹 “ 𝑧))
20		nrmsep3 23297	. . . . 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 1374	. . . 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 774	. . . . . 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 770	. . . . . 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 23676	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐽) → (𝐹 “ 𝑢) ∈ (KQ‘𝐽))
25	22, 23, 24	syl2anc 584	. . . . 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 780	. . . . . 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 23667	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
28		fnfun 6590	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
29	22, 27, 28	3syl 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)))) → Fun 𝐹)
30	13	adantr 480	. . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
32	31	cldss 22971	. . . . . . . . 9 ⊢ (𝑤 ∈ (Clsd‘(KQ‘𝐽)) → 𝑤 ⊆ ∪ (KQ‘𝐽))
33	30, 32	syl 17	. . . . . . . 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 22856	. . . . . . . . 9 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
35	22, 2, 34	3syl 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)))) → ran 𝐹 = ∪ (KQ‘𝐽))
36	33, 35	sseqtrrd 3969	. . . . . . 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 6572	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ ran 𝐹) → ((^◡𝐹 “ 𝑤) ⊆ 𝑢 → 𝑤 ⊆ (𝐹 “ 𝑢)))
38	29, 36, 37	syl2anc 584	. . . . . 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 15	. . . . 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 22855	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
41	22, 40	syl 17	. . . . . . . . 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 4892	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽)
43	42	ad2antrl 728	. . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
45	44	clscld 22989	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46	41, 43, 45	syl2anc 584	. . . . . . . 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 23677	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)))
48	22, 46, 47	syl2anc 584	. . . . . . 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 22998	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
50	41, 43, 49	syl2anc 584	. . . . . . . 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 6059	. . . . . . . 8 ⊢ (𝑢 ⊆ ((cls‘𝐽)‘𝑢) → (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
52	50, 51	syl 17	. . . . . . 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 23014	. . . . . . 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 584	. . . . . 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 781	. . . . . . 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 23001	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)
57	41, 43, 56	syl2anc 584	. . . . . . . . 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 6593	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
59	22, 27, 58	3syl 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)))) → dom 𝐹 = 𝑋)
60		toponuni 22856	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
61	22, 60	syl 17	. . . . . . . . . 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 2769	. . . . . . . . 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 3969	. . . . . . . 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 6997	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (^◡𝐹 “ 𝑧)))
65	29, 63, 64	syl2anc 584	. . . . . . 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 257	. . . . . 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 3942	. . . . 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 3958	. . . . . . 7 ⊢ (𝑚 = (𝐹 “ 𝑢) → (𝑤 ⊆ 𝑚 ↔ 𝑤 ⊆ (𝐹 “ 𝑢)))
69		fveq2 6832	. . . . . . . 8 ⊢ (𝑚 = (𝐹 “ 𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)))
70	69	sseq1d 3963	. . . . . . 7 ⊢ (𝑚 = (𝐹 “ 𝑢) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧))
71	68, 70	anbi12d 632	. . . . . 6 ⊢ (𝑚 = (𝐹 “ 𝑢) → ((𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹 “ 𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)))
72	71	rspcev 3574	. . . . 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 836	. . . 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 3141	. . 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 3177	. 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 23277	. 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 583	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 {crab 3397 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4552 ∪ cuni 4861 ↦ cmpt 5177 ^◡ccnv 5621 dom cdm 5622 ran crn 5623 “ cima 5625 Fun wfun 6484 Fn wfn 6485 ‘cfv 6490 (class class class)co 7356 Topctop 22835 TopOnctopon 22852 Clsdccld 22958 clsccl 22960 Cn ccn 23166 Nrmcnrm 23252 KQckq 23635

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-map 8763 df-qtop 17426 df-top 22836 df-topon 22853 df-cld 22961 df-cls 22963 df-cn 23169 df-nrm 23259 df-kq 23636

This theorem is referenced by: kqnrm 23694

W3C validator