kqnrmlem2 - Metamath Proof Explorer

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Theorem kqnrmlem2 22355

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 21524	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 483	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Top)
3		simplr 767	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (KQ‘𝐽) ∈ Nrm)
4		simpll 765	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝐽 ∈ (TopOn‘𝑋))
5		simprl 769	. . . . . 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 22345	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
8	4, 5, 7	syl2anc 586	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
9		simprr 771	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))
10	9	elin1d 4178	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘𝐽))
11	6	kqcld 22346	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
12	4, 10, 11	syl2anc 586	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
13	9	elin2d 4179	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
14		elpwi 4551	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧)
15		imass2 5968	. . . . . 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 21966	. . . . 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 1368	. . . 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 773	. . . . . . 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 22339	. . . . . . 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 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)))) → 𝑚 ∈ (KQ‘𝐽))
23		cnima 21876	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑚 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑚) ∈ 𝐽)
24	21, 22, 23	syl2anc 586	. . . . 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 779	. . . . . 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 22336	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27		fnfun 6456	. . . . . . . 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 483	. . . . . . . . 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 2824	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
31	30	cldss 21640	. . . . . . . . 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 6458	. . . . . . . . . 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 21525	. . . . . . . . . 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 2859	. . . . . . . 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 4011	. . . . . . 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 6827	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ 𝑚 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
40	28, 38, 39	syl2anc 586	. . . . . 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 234	. . . . 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 22338	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
43		topontop 21524	. . . . . . . . . 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 4871	. . . . . . . . . 10 ⊢ (𝑚 ∈ (KQ‘𝐽) → 𝑚 ⊆ ∪ (KQ‘𝐽))
46	45	ad2antrl 726	. . . . . . . . 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 2824	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
48	47	clscld 21658	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
49	44, 46, 48	syl2anc 586	. . . . . . . 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 21879	. . . . . . . 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 586	. . . . . . 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 21667	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
53	44, 46, 52	syl2anc 586	. . . . . . . 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 5968	. . . . . . . 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 21683	. . . . . . 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 586	. . . . . 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 780	. . . . . . . 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 5968	. . . . . . . 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 483	. . . . . . . 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 22342	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
63	19, 61, 62	syl2anc 586	. . . . . . 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 4010	. . . . . 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 3980	. . . . 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 3996	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (𝑤 ⊆ 𝑢 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
67		fveq2 6673	. . . . . . . 8 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((cls‘𝐽)‘𝑢) = ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)))
68	67	sseq1d 4001	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (((cls‘𝐽)‘𝑢) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧))
69	66, 68	anbi12d 632	. . . . . 6 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧) ↔ (𝑤 ⊆ (^◡𝐹 “ 𝑚) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)))
70	69	rspcev 3626	. . . . 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 834	. . . 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 3294	. . 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 3194	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
74		isnrm 21946	. 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 585	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 {crab 3145 ∩ cin 3938 ⊆ wss 3939 𝒫 cpw 4542 ∪ cuni 4841 ↦ cmpt 5149 ^◡ccnv 5557 dom cdm 5558 ran crn 5559 “ cima 5561 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 (class class class)co 7159 Topctop 21504 TopOnctopon 21521 Clsdccld 21627 clsccl 21629 Cn ccn 21835 Nrmcnrm 21921 KQckq 22304

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-qtop 16783 df-top 21505 df-topon 21522 df-cld 21630 df-cls 21632 df-cn 21838 df-nrm 21928 df-kq 22305

This theorem is referenced by: kqnrm 22363

W3C validator