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Theorem kqnrmlem2 22071

Description: If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)

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

**Assertion**
Ref	Expression
kqnrmlem2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)

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

Proof of Theorem kqnrmlem2 Dummy variables 𝑚 𝑤 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 21240	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 473	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Top)
3		simplr 757	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (KQ‘𝐽) ∈ Nrm)
4		simpll 755	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝐽 ∈ (TopOn‘𝑋))
5		simprl 759	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑧 ∈ 𝐽)
6		kqval.2	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
7	6	kqopn 22061	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
8	4, 5, 7	syl2anc 576	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
9		simprr 761	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))
10	9	elin1d 4057	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘𝐽))
11	6	kqcld 22062	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
12	4, 10, 11	syl2anc 576	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
13	9	elin2d 4058	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
14		elpwi 4426	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧)
15		imass2 5802	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))
16	13, 14, 15	3syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))
17		nrmsep3 21682	. . . . 5 ⊢ (((KQ‘𝐽) ∈ Nrm ∧ ((𝐹 “ 𝑧) ∈ (KQ‘𝐽) ∧ (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))
18	3, 8, 12, 16, 17	syl13anc 1353	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))
19		simplll 763	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
20	6	kqid 22055	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
21	19, 20	syl 17	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22		simprl 759	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑚 ∈ (KQ‘𝐽))
23		cnima 21592	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑚 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑚) ∈ 𝐽)
24	21, 22, 23	syl2anc 576	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ 𝑚) ∈ 𝐽)
25		simprrl 769	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (𝐹 “ 𝑤) ⊆ 𝑚)
26	6	kqffn 22052	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27		fnfun 6283	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
28	19, 26, 27	3syl 18	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → Fun 𝐹)
29	10	adantr 473	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ∈ (Clsd‘𝐽))
30		eqid 2771	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
31	30	cldss 21356	. . . . . . . . 9 ⊢ (𝑤 ∈ (Clsd‘𝐽) → 𝑤 ⊆ ∪ 𝐽)
32	29, 31	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ⊆ ∪ 𝐽)
33		fndm 6285	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
34	19, 26, 33	3syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → dom 𝐹 = 𝑋)
35		toponuni 21241	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
36	19, 35	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑋 = ∪ 𝐽)
37	34, 36	eqtrd 2807	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → dom 𝐹 = ∪ 𝐽)
38	32, 37	sseqtr4d 3891	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ⊆ dom 𝐹)
39		funimass3 6647	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ 𝑚 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
40	28, 38, 39	syl2anc 576	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((𝐹 “ 𝑤) ⊆ 𝑚 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
41	25, 40	mpbid 224	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ⊆ (^◡𝐹 “ 𝑚))
42	6	kqtopon 22054	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
43		topontop 21240	. . . . . . . . . 10 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
44	19, 42, 43	3syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (KQ‘𝐽) ∈ Top)
45		elssuni 4737	. . . . . . . . . 10 ⊢ (𝑚 ∈ (KQ‘𝐽) → 𝑚 ⊆ ∪ (KQ‘𝐽))
46	45	ad2antrl 716	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑚 ⊆ ∪ (KQ‘𝐽))
47		eqid 2771	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
48	47	clscld 21374	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
49	44, 46, 48	syl2anc 576	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
50		cnclima 21595	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
51	21, 49, 50	syl2anc 576	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
52	47	sscls 21383	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
53	44, 46, 52	syl2anc 576	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
54		imass2 5802	. . . . . . . 8 ⊢ (𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚) → (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
55	53, 54	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
56	30	clsss2 21399	. . . . . . 7 ⊢ (((^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽) ∧ (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
57	51, 55, 56	syl2anc 576	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
58		simprrr 770	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧))
59		imass2 5802	. . . . . . . 8 ⊢ (((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
60	58, 59	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
61	5	adantr 473	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑧 ∈ 𝐽)
62	6	kqsat 22058	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
63	19, 61, 62	syl2anc 576	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
64	60, 63	sseqtrd 3890	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ 𝑧)
65	57, 64	sstrd 3861	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)
66		sseq2 3876	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (𝑤 ⊆ 𝑢 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
67		fveq2 6496	. . . . . . . 8 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((cls‘𝐽)‘𝑢) = ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)))
68	67	sseq1d 3881	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (((cls‘𝐽)‘𝑢) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧))
69	66, 68	anbi12d 622	. . . . . 6 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧) ↔ (𝑤 ⊆ (^◡𝐹 “ 𝑚) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)))
70	69	rspcev 3528	. . . . 5 ⊢ (((^◡𝐹 “ 𝑚) ∈ 𝐽 ∧ (𝑤 ⊆ (^◡𝐹 “ 𝑚) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)) → ∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
71	24, 41, 65, 70	syl12anc 825	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
72	18, 71	rexlimddv 3229	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
73	72	ralrimivva 3134	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
74		isnrm 21662	. 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧)))
75	2, 73, 74	sylanbrc 575	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ∃wrex 3082 {crab 3085 ∩ cin 3821 ⊆ wss 3822 𝒫 cpw 4416 ∪ cuni 4708 ↦ cmpt 5004 ^◡ccnv 5402 dom cdm 5403 ran crn 5404 “ cima 5406 Fun wfun 6179 Fn wfn 6180 ‘cfv 6185 (class class class)co 6974 Topctop 21220 TopOnctopon 21237 Clsdccld 21343 clsccl 21345 Cn ccn 21551 Nrmcnrm 21637 KQckq 22020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-map 8206 df-qtop 16634 df-top 21221 df-topon 21238 df-cld 21346 df-cls 21348 df-cn 21554 df-nrm 21644 df-kq 22021

This theorem is referenced by: kqnrm 22079

W3C validator