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Theorem kqreglem2 23605

Description: If the Kolmogorov quotient of a space is regular 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
kqreglem2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)

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

Proof of Theorem kqreglem2 Dummy variables 𝑚 𝑛 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 22776	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top)
3		simplr 768	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (KQ‘𝐽) ∈ Reg)
4		simpll 766	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝐽 ∈ (TopOn‘𝑋))
5		simprl 770	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑧 ∈ 𝐽)
6		kqval.2	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
7	6	kqopn 23597	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
8	4, 5, 7	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
9		simprr 772	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑧)
10		toponss 22790	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
11	4, 5, 10	syl2anc 584	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑧 ⊆ 𝑋)
12	11, 9	sseldd 3944	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑋)
13	6	kqfvima 23593	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)))
14	4, 5, 12, 13	syl3anc 1373	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝑤 ∈ 𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)))
15	9, 14	mpbid 232	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑧))
16		regsep 23197	. . . . 5 ⊢ (((KQ‘𝐽) ∈ Reg ∧ (𝐹 “ 𝑧) ∈ (KQ‘𝐽) ∧ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))
17	3, 8, 15, 16	syl3anc 1373	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))
18	4	adantr 480	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
19	6	kqid 23591	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
20	18, 19	syl 17	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
21		simprl 770	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑛 ∈ (KQ‘𝐽))
22		cnima 23128	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑛 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑛) ∈ 𝐽)
23	20, 21, 22	syl2anc 584	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ 𝑛) ∈ 𝐽)
24	12	adantr 480	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ∈ 𝑋)
25		simprrl 780	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (𝐹‘𝑤) ∈ 𝑛)
26	6	kqffn 23588	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27		elpreima 7012	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → (𝑤 ∈ (^◡𝐹 “ 𝑛) ↔ (𝑤 ∈ 𝑋 ∧ (𝐹‘𝑤) ∈ 𝑛)))
28	18, 26, 27	3syl 18	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (𝑤 ∈ (^◡𝐹 “ 𝑛) ↔ (𝑤 ∈ 𝑋 ∧ (𝐹‘𝑤) ∈ 𝑛)))
29	24, 25, 28	mpbir2and 713	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ∈ (^◡𝐹 “ 𝑛))
30	6	kqtopon 23590	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
31		topontop 22776	. . . . . . . . . 10 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
32	18, 30, 31	3syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (KQ‘𝐽) ∈ Top)
33		elssuni 4897	. . . . . . . . . 10 ⊢ (𝑛 ∈ (KQ‘𝐽) → 𝑛 ⊆ ∪ (KQ‘𝐽))
34	33	ad2antrl 728	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑛 ⊆ ∪ (KQ‘𝐽))
35		eqid 2729	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
36	35	clscld 22910	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑛 ⊆ ∪ (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽)))
37	32, 34, 36	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽)))
38		cnclima 23131	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽))
39	20, 37, 38	syl2anc 584	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽))
40	35	sscls 22919	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑛 ⊆ ∪ (KQ‘𝐽)) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛))
41	32, 34, 40	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛))
42		imass2 6062	. . . . . . . 8 ⊢ (𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛) → (^◡𝐹 “ 𝑛) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
43	41, 42	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ 𝑛) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
44		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
45	44	clsss2 22935	. . . . . . 7 ⊢ (((^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽) ∧ (^◡𝐹 “ 𝑛) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
46	39, 43, 45	syl2anc 584	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)))
47		simprrr 781	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧))
48		imass2 6062	. . . . . . . 8 ⊢ (((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
49	47, 48	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
50	5	adantr 480	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑧 ∈ 𝐽)
51	6	kqsat 23594	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
52	18, 50, 51	syl2anc 584	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
53	49, 52	sseqtrd 3980	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ 𝑧)
54	46, 53	sstrd 3954	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ 𝑧)
55		eleq2 2817	. . . . . . 7 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → (𝑤 ∈ 𝑚 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑛)))
56		fveq2 6840	. . . . . . . 8 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)))
57	56	sseq1d 3975	. . . . . . 7 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → (((cls‘𝐽)‘𝑚) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ 𝑧))
58	55, 57	anbi12d 632	. . . . . 6 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → ((𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧) ↔ (𝑤 ∈ (^◡𝐹 “ 𝑛) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ 𝑧)))
59	58	rspcev 3585	. . . . 5 ⊢ (((^◡𝐹 “ 𝑛) ∈ 𝐽 ∧ (𝑤 ∈ (^◡𝐹 “ 𝑛) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ 𝑧)) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
60	23, 29, 54, 59	syl12anc 836	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
61	17, 60	rexlimddv 3140	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
62	61	ralrimivva 3178	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ 𝑧 ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
63		isreg 23195	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ 𝑧 ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)))
64	2, 62, 63	sylanbrc 583	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3402 ⊆ wss 3911 ∪ cuni 4867 ↦ cmpt 5183 ^◡ccnv 5630 ran crn 5632 “ cima 5634 Fn wfn 6494 ‘cfv 6499 (class class class)co 7369 Topctop 22756 TopOnctopon 22773 Clsdccld 22879 clsccl 22881 Cn ccn 23087 Regcreg 23172 KQckq 23556

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

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 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-map 8778 df-qtop 17446 df-top 22757 df-topon 22774 df-cld 22882 df-cls 22884 df-cn 23090 df-reg 23179 df-kq 23557

This theorem is referenced by: kqreg 23614

W3C validator