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Theorem kqreglem1 23639

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 23625	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3	2	adantr 480	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4		topontop 22809	. . 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 22823	. . . . . . . 8 ⊢ (((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
7	3, 6	sylan 579	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
8	7	sselda 3979	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ ran 𝐹)
9	1	kqffn 23623	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
10	9	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝐹 Fn 𝑋)
11		fvelrnb 6954	. . . . . . 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 775	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg)
15	1	kqid 23626	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
16	15	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
17		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽))
18		cnima 23163	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑎) ∈ 𝐽)
19	16, 17, 18	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → (^◡𝐹 “ 𝑎) ∈ 𝐽)
20	9	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
21	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
22		elpreima 7062	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑧 ∈ (^◡𝐹 “ 𝑎))
25		regsep 23232	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Reg ∧ (^◡𝐹 “ 𝑎) ∈ 𝐽 ∧ 𝑧 ∈ (^◡𝐹 “ 𝑎)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))
26	14, 19, 24, 25	syl3anc 1369	. . . . . . . . . . . 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 782	. . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . 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 23632	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
30	27, 28, 29	syl2anc 583	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 23628	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑤 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
34	27, 28, 32, 33	syl3anc 1369	. . . . . . . . . . . . . 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 22809	. . . . . . . . . . . . . . . . . 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 4936	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
39	38	ad2antrl 727	. . . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	clscld 22945	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
42	37, 39, 41	syl2anc 583	. . . . . . . . . . . . . . . 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 23633	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
44	27, 42, 43	syl2anc 583	. . . . . . . . . . . . . . 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 22954	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
46	37, 39, 45	syl2anc 583	. . . . . . . . . . . . . . . 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 6101	. . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . 16 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
50	49	clsss2 22970	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 6649	. . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . 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 6631	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
57	54, 55, 56	syl2anc 583	. . . . . . . . . . . . . 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 3989	. . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((𝐹‘𝑧) ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
60		fveq2 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)))
61	60	sseq1d 4010	. . . . . . . . . . . . . . 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 3608	. . . . . . . . . . . . 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 836	. . . . . . . . . . . 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 3157	. . . . . . . . . . 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 456	. . . . . . . . . 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 2817	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑎 ↔ 𝑏 ∈ 𝑎))
68		eleq1 2817	. . . . . . . . . . . . 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 3174	. . . . . . . . . . 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 344	. . . . . . . . . 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 406	. . . . . . 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 651	. . . . . 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 3151	. . . . 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 466	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑎 ∈ (KQ‘𝐽) ∧ 𝑏 ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
79	78	ralrimivva 3196	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏 ∈ 𝑎 ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
80		isreg 23230	. 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 582	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 {crab 3428 ⊆ wss 3945 ∪ cuni 4904 ↦ cmpt 5226 ^◡ccnv 5672 ran crn 5674 “ cima 5676 Fun wfun 6537 Fn wfn 6538 ‘cfv 6543 (class class class)co 7415 Topctop 22789 TopOnctopon 22806 Clsdccld 22914 clsccl 22916 Cn ccn 23122 Regcreg 23207 KQckq 23591

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-map 8841 df-qtop 17483 df-top 22790 df-topon 22807 df-cld 22917 df-cls 22919 df-cn 23125 df-reg 23214 df-kq 23592

This theorem is referenced by: kqreg 23649

W3C validator