kqreglem2 - Metamath Proof Explorer

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Theorem kqreglem2 23750

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 22919	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top)
3		simplr 769	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (KQ‘𝐽) ∈ Reg)
4		simpll 767	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝐽 ∈ (TopOn‘𝑋))
5		simprl 771	. . . . . 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 23742	. . . . . 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 773	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑧)
10		toponss 22933	. . . . . . . . 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 3984	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑋)
13	6	kqfvima 23738	. . . . . . 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 23342	. . . . 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 23736	. . . . . . 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 771	. . . . . 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 23273	. . . . . 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 781	. . . . . 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 23733	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27		elpreima 7078	. . . . . . 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 23735	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
31		topontop 22919	. . . . . . . . . 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 4937	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
36	35	clscld 23055	. . . . . . . . 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 23276	. . . . . . . 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 23064	. . . . . . . . 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 6120	. . . . . . . 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 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
45	44	clsss2 23080	. . . . . . 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 782	. . . . . . . 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 6120	. . . . . . . 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 23739	. . . . . . . 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 4020	. . . . . 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 3994	. . . . 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 2830	. . . . . . 7 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → (𝑤 ∈ 𝑚 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑛)))
56		fveq2 6906	. . . . . . . 8 ⊢ (𝑚 = (^◡𝐹 “ 𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(^◡𝐹 “ 𝑛)))
57	56	sseq1d 4015	. . . . . . 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 3622	. . . . 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 837	. . . 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 3161	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
62	61	ralrimivva 3202	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ 𝑧 ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))
63		isreg 23340	. 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 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∪ cuni 4907 ↦ cmpt 5225 ^◡ccnv 5684 ran crn 5686 “ cima 5688 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 Topctop 22899 TopOnctopon 22916 Clsdccld 23024 clsccl 23026 Cn ccn 23232 Regcreg 23317 KQckq 23701

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-qtop 17552 df-top 22900 df-topon 22917 df-cld 23027 df-cls 23029 df-cn 23235 df-reg 23324 df-kq 23702

This theorem is referenced by: kqreg 23759

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