kqnrmlem2 - Metamath Proof Explorer

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Theorem kqnrmlem2 23901

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 23070	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 485	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Top)
3		simplr 780	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (KQ‘𝐽) ∈ Nrm)
4		simpll 778	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝐽 ∈ (TopOn‘𝑋))
5		simprl 782	. . . . . 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 23891	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
8	4, 5, 7	syl2anc 595	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
9		simprr 784	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))
10	9	elin1d 4157	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘𝐽))
11	6	kqcld 23892	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
12	4, 10, 11	syl2anc 595	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
13	9	elin2d 4158	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
14		elpwi 4569	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧)
15		imass2 6104	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))
16	13, 14, 15	3syl 19	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))
17		nrmsep3 23512	. . . . 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 1399	. . . 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 786	. . . . . . 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 23885	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
21	19, 20	syl 18	. . . . . 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 782	. . . . . 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 23422	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑚 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑚) ∈ 𝐽)
24	21, 22, 23	syl2anc 595	. . . . 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 792	. . . . . 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 23882	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27		fnfun 6635	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
28	19, 26, 27	3syl 19	. . . . . . 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 485	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
31	30	cldss 23186	. . . . . . . . 9 ⊢ (𝑤 ∈ (Clsd‘𝐽) → 𝑤 ⊆ ∪ 𝐽)
32	29, 31	syl 18	. . . . . . . 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 6638	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
34	19, 26, 33	3syl 19	. . . . . . . . 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 23071	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
36	19, 35	syl 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)))) → 𝑋 = ∪ 𝐽)
37	34, 36	eqtrd 2798	. . . . . . . 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	sseqtrrd 3974	. . . . . . 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 7049	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ 𝑚 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
40	28, 38, 39	syl2anc 595	. . . . . 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 235	. . . . 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 23884	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
43		topontop 23070	. . . . . . . . . 10 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
44	19, 42, 43	3syl 19	. . . . . . . . 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 4904	. . . . . . . . . 10 ⊢ (𝑚 ∈ (KQ‘𝐽) → 𝑚 ⊆ ∪ (KQ‘𝐽))
46	45	ad2antrl 740	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
48	47	clscld 23204	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
49	44, 46, 48	syl2anc 595	. . . . . . . 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 23425	. . . . . . . 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 595	. . . . . . 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 23213	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
53	44, 46, 52	syl2anc 595	. . . . . . . 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 6104	. . . . . . . 8 ⊢ (𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚) → (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
55	53, 54	syl 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)))) → (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
56	30	clsss2 23229	. . . . . . 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 595	. . . . . 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 793	. . . . . . . 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 6104	. . . . . . . 8 ⊢ (((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
60	58, 59	syl 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)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
61	5	adantr 485	. . . . . . . 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 23888	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
63	19, 61, 62	syl2anc 595	. . . . . . 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 3973	. . . . . 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 3947	. . . . 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 3963	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (𝑤 ⊆ 𝑢 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
67		fveq2 6881	. . . . . . . 8 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((cls‘𝐽)‘𝑢) = ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)))
68	67	sseq1d 3968	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (((cls‘𝐽)‘𝑢) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧))
69	66, 68	anbi12d 643	. . . . . 6 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧) ↔ (𝑤 ⊆ (^◡𝐹 “ 𝑚) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)))
70	69	rspcev 3581	. . . . 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 849	. . . 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 3172	. . 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 3208	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
74		isnrm 23492	. 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 594	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 {crab 3416 ∩ cin 3904 ⊆ wss 3905 𝒫 cpw 4562 ∪ cuni 4872 ↦ cmpt 5192 ^◡ccnv 5660 dom cdm 5661 ran crn 5662 “ cima 5664 Fun wfun 6530 Fn wfn 6531 ‘cfv 6536 (class class class)co 7410 Topctop 23050 TopOnctopon 23067 Clsdccld 23173 clsccl 23175 Cn ccn 23381 Nrmcnrm 23467 KQckq 23850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-map 8822 df-qtop 17556 df-top 23051 df-topon 23068 df-cld 23176 df-cls 23178 df-cn 23384 df-nrm 23474 df-kq 23851

This theorem is referenced by: kqnrm 23909

W3C validator