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Theorem kqnrmlem1 23628

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 23612	. . . 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 22798	. . 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 23613	. . . . . . 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 23150	. . . . . 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 4155	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
14		cnclima 23153	. . . . . 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 4156	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
17		elpwi 4558	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧)
18		imass2 6053	. . . . . 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 23240	. . . . 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 23619	. . . . . 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 23610	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
28		fnfun 6582	. . . . . . . 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 2729	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
32	31	cldss 22914	. . . . . . . . 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 22799	. . . . . . . . 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 3973	. . . . . . 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 6564	. . . . . . 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 22798	. . . . . . . . . 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 4888	. . . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
45	44	clscld 22932	. . . . . . . . 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 23620	. . . . . . . 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 22941	. . . . . . . . 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 6053	. . . . . . . 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 22957	. . . . . . 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 22944	. . . . . . . . . 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 6585	. . . . . . . . . . 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 22799	. . . . . . . . . . 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 2764	. . . . . . . . 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 3973	. . . . . . . 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 6988	. . . . . . . 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 3946	. . . . 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 3962	. . . . . . 7 ⊢ (𝑚 = (𝐹 “ 𝑢) → (𝑤 ⊆ 𝑚 ↔ 𝑤 ⊆ (𝐹 “ 𝑢)))
69		fveq2 6822	. . . . . . . 8 ⊢ (𝑚 = (𝐹 “ 𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)))
70	69	sseq1d 3967	. . . . . . 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 3577	. . . . 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 3136	. . 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 3172	. 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 23220	. 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 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3394 ∩ cin 3902 ⊆ wss 3903 𝒫 cpw 4551 ∪ cuni 4858 ↦ cmpt 5173 ^◡ccnv 5618 dom cdm 5619 ran crn 5620 “ cima 5622 Fun wfun 6476 Fn wfn 6477 ‘cfv 6482 (class class class)co 7349 Topctop 22778 TopOnctopon 22795 Clsdccld 22901 clsccl 22903 Cn ccn 23109 Nrmcnrm 23195 KQckq 23578

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-map 8755 df-qtop 17411 df-top 22779 df-topon 22796 df-cld 22904 df-cls 22906 df-cn 23112 df-nrm 23202 df-kq 23579

This theorem is referenced by: kqnrm 23637

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