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Theorem kqreglem1 23706

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 23692	. . . 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 22878	. . 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 22892	. . . . . . . 8 ⊢ (((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
7	3, 6	sylan 581	. . . . . . 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 23690	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
10	9	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝐹 Fn 𝑋)
11		fvelrnb 6900	. . . . . . 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 232	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏)
14		simpllr 776	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg)
15	1	kqid 23693	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
16	15	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
17		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽))
18		cnima 23230	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑎) ∈ 𝐽)
19	16, 17, 18	syl2anc 585	. . . . . . . . . . . . 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 7010	. . . . . . . . . . . . . . 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 23299	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Reg ∧ (^◡𝐹 “ 𝑎) ∈ 𝐽 ∧ 𝑧 ∈ (^◡𝐹 “ 𝑎)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))
26	14, 19, 24, 25	syl3anc 1374	. . . . . . . . . . . 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 783	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 23699	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
30	27, 28, 29	syl2anc 585	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . 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 23695	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑤 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
34	27, 28, 32, 33	syl3anc 1374	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 22878	. . . . . . . . . . . . . . . . . 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 4881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
39	38	ad2antrl 729	. . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	clscld 23012	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
42	37, 39, 41	syl2anc 585	. . . . . . . . . . . . . . . 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 23700	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
44	27, 42, 43	syl2anc 585	. . . . . . . . . . . . . . 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 23021	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
46	37, 39, 45	syl2anc 585	. . . . . . . . . . . . . . . 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 6067	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 16 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
50	49	clsss2 23037	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . 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 6598	. . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . 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 6581	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
57	54, 55, 56	syl2anc 585	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((𝐹‘𝑧) ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
60		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)))
61	60	sseq1d 3953	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹 “ 𝑤) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎))
62	59, 61	anbi12d 633	. . . . . . . . . . . . . 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 3564	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 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 3144	. . . . . . . . . . 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 2824	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑎 ↔ 𝑏 ∈ 𝑎))
68		eleq1 2824	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑚 ↔ 𝑏 ∈ 𝑚))
69	68	anbi1d 632	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) = 𝑏 → (((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ (𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
70	69	rexbidv 3161	. . . . . . . . . . 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 245	. . . . . . . . 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 653	. . . . . 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 3138	. . . . 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 3180	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏 ∈ 𝑎 ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
80		isreg 23297	. 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 584	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 {crab 3389 ⊆ wss 3889 ∪ cuni 4850 ↦ cmpt 5166 ^◡ccnv 5630 ran crn 5632 “ cima 5634 Fun wfun 6492 Fn wfn 6493 ‘cfv 6498 (class class class)co 7367 Topctop 22858 TopOnctopon 22875 Clsdccld 22981 clsccl 22983 Cn ccn 23189 Regcreg 23274 KQckq 23658

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 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 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-map 8775 df-qtop 17471 df-top 22859 df-topon 22876 df-cld 22984 df-cls 22986 df-cn 23192 df-reg 23281 df-kq 23659

This theorem is referenced by: kqreg 23716

W3C validator