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Theorem kqreglem1 22892

Description: A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)

**Hypothesis**
Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

**Assertion**
Ref	Expression
kqreglem1	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)

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

Proof of Theorem kqreglem1 Dummy variables 𝑚 𝑤 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqtopon 22878	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3	2	adantr 481	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4		topontop 22062	. . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
5	3, 4	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Top)
6		toponss 22076	. . . . . . . 8 ⊢ (((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
7	3, 6	sylan 580	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
8	7	sselda 3921	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ ran 𝐹)
9	1	kqffn 22876	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
10	9	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝐹 Fn 𝑋)
11		fvelrnb 6830	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏))
12	10, 11	syl 17	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏))
13	8, 12	mpbid 231	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏)
14		simpllr 773	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg)
15	1	kqid 22879	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
16	15	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
17		simplr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽))
18		cnima 22416	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑎) ∈ 𝐽)
19	16, 17, 18	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → (^◡𝐹 “ 𝑎) ∈ 𝐽)
20	9	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
21	20	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
22		elpreima 6935	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝑋 → (𝑧 ∈ (^◡𝐹 “ 𝑎) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)))
23	21, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → (𝑧 ∈ (^◡𝐹 “ 𝑎) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)))
24	23	biimpar 478	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑧 ∈ (^◡𝐹 “ 𝑎))
25		regsep 22485	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Reg ∧ (^◡𝐹 “ 𝑎) ∈ 𝐽 ∧ 𝑧 ∈ (^◡𝐹 “ 𝑎)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))
26	14, 19, 24, 25	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))
27		simp-4l 780	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝐽 ∈ (TopOn‘𝑋))
28		simprl 768	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝑤 ∈ 𝐽)
29	1	kqopn 22885	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
30	27, 28, 29	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
31		simprrl 778	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝑧 ∈ 𝑤)
32		simplrl 774	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝑧 ∈ 𝑋)
33	1	kqfvima 22881	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑤 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
34	27, 28, 32, 33	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → (𝑧 ∈ 𝑤 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
35	31, 34	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → (𝐹‘𝑧) ∈ (𝐹 “ 𝑤))
36		topontop 22062	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
37	27, 36	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝐽 ∈ Top)
38		elssuni 4871	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
39	38	ad2antrl 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝑤 ⊆ ∪ 𝐽)
40		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	clscld 22198	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
42	37, 39, 41	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
43	1	kqcld 22886	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
44	27, 42, 43	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
45	40	sscls 22207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
46	37, 39, 45	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
47		imass2 6010	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ⊆ ((cls‘𝐽)‘𝑤) → (𝐹 “ 𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
48	46, 47	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → (𝐹 “ 𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
49		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
50	49	clsss2 22223	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹 “ 𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
51	44, 48, 50	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
52	20	ad3antrrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → 𝐹 Fn 𝑋)
53		fnfun 6533	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
54	52, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → Fun 𝐹)
55		simprrr 779	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎))
56		funimass2 6517	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
57	54, 55, 56	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
58	51, 57	sstrd 3931	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎)
59		eleq2 2827	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((𝐹‘𝑧) ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
60		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)))
61	60	sseq1d 3952	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹 “ 𝑤) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎))
62	59, 61	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝐹 “ 𝑤) → (((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎)))
63	62	rspcev 3561	. . . . . . . . . . . . 13 ⊢ (((𝐹 “ 𝑤) ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
64	30, 35, 58, 63	syl12anc 834	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
65	26, 64	rexlimddv 3220	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
66	65	expr 457	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
67		eleq1 2826	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑎 ↔ 𝑏 ∈ 𝑎))
68		eleq1 2826	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑚 ↔ 𝑏 ∈ 𝑚))
69	68	anbi1d 630	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) = 𝑏 → (((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ (𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
70	69	rexbidv 3226	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑏 → (∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
71	67, 70	imbi12d 345	. . . . . . . . . 10 ⊢ ((𝐹‘𝑧) = 𝑏 → (((𝐹‘𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) ↔ (𝑏 ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
72	66, 71	syl5ibcom 244	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = 𝑏 → (𝑏 ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
73	72	com23 86	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑏 ∈ 𝑎 → ((𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
74	73	imp 407	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑎) → ((𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
75	74	an32s 649	. . . . . 6 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
76	75	rexlimdva 3213	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → (∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
77	13, 76	mpd 15	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
78	77	anasss 467	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑎 ∈ (KQ‘𝐽) ∧ 𝑏 ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
79	78	ralrimivva 3123	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏 ∈ 𝑎 ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
80		isreg 22483	. 2 ⊢ ((KQ‘𝐽) ∈ Reg ↔ ((KQ‘𝐽) ∈ Top ∧ ∀𝑎 ∈ (KQ‘𝐽)∀𝑏 ∈ 𝑎 ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
81	5, 79, 80	sylanbrc 583	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 ⊆ wss 3887 ∪ cuni 4839 ↦ cmpt 5157 ^◡ccnv 5588 ran crn 5590 “ cima 5592 Fun wfun 6427 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 Topctop 22042 TopOnctopon 22059 Clsdccld 22167 clsccl 22169 Cn ccn 22375 Regcreg 22460 KQckq 22844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-qtop 17218 df-top 22043 df-topon 22060 df-cld 22170 df-cls 22172 df-cn 22378 df-reg 22467 df-kq 22845

This theorem is referenced by: kqreg 22902

W3C validator