kqreglem2 - Metamath Proof Explorer

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Theorem kqreglem2 22891

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 22060	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 481	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top)
3		simplr 766	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (KQ‘𝐽) ∈ Reg)
4		simpll 764	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝐽 ∈ (TopOn‘𝑋))
5		simprl 768	. . . . . 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 22883	. . . . . 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 770	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑧)
10		toponss 22074	. . . . . . . . 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 3927	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑋)
13	6	kqfvima 22879	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)))
14	4, 5, 12, 13	syl3anc 1370	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝑤 ∈ 𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)))
15	9, 14	mpbid 231	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑧))
16		regsep 22483	. . . . 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 1370	. . . 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 481	. . . . . . 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 22877	. . . . . . 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 768	. . . . . 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 22414	. . . . . 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 481	. . . . . 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 778	. . . . . 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 22874	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27		elpreima 6932	. . . . . . 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 710	. . . . 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 22876	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
31		topontop 22060	. . . . . . . . . 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 4877	. . . . . . . . . 10 ⊢ (𝑛 ∈ (KQ‘𝐽) → 𝑛 ⊆ ∪ (KQ‘𝐽))
34	33	ad2antrl 725	. . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
36	35	clscld 22196	. . . . . . . . 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 22417	. . . . . . . 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 22205	. . . . . . . . 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 6009	. . . . . . . 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 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
45	44	clsss2 22221	. . . . . . 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 779	. . . . . . . 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 6009	. . . . . . . 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 481	. . . . . . . 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 22880	. . . . . . . 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 3966	. . . . . 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 3936	. . . . 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 2829	. . . . . . 7 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → (𝑤 ∈ 𝑚 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑛)))
56		fveq2 6771	. . . . . . . 8 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)))
57	56	sseq1d 3957	. . . . . . 7 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → (((cls‘𝐽)‘𝑚) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ 𝑧))
58	55, 57	anbi12d 631	. . . . . 6 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → ((𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧) ↔ (𝑤 ∈ (^◡𝐹 “ 𝑛) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)) ⊆ 𝑧)))
59	58	rspcev 3561	. . . . 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 834	. . . 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 3222	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
62	61	ralrimivva 3117	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ 𝑧 ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
63		isreg 22481	. 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 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 {crab 3070 ⊆ wss 3892 ∪ cuni 4845 ↦ cmpt 5162 ^◡ccnv 5589 ran crn 5591 “ cima 5593 Fn wfn 6427 ‘cfv 6432 (class class class)co 7271 Topctop 22040 TopOnctopon 22057 Clsdccld 22165 clsccl 22167 Cn ccn 22373 Regcreg 22458 KQckq 22842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-map 8600 df-qtop 17216 df-top 22041 df-topon 22058 df-cld 22168 df-cls 22170 df-cn 22376 df-reg 22465 df-kq 22843

This theorem is referenced by: kqreg 22900

W3C validator