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Theorem kqreglem1 22351

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 22337	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3	2	adantr 483	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4		topontop 21523	. . 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 21537	. . . . . . . 8 ⊢ (((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
7	3, 6	sylan 582	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
8	7	sselda 3969	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ ran 𝐹)
9	1	kqffn 22335	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
10	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝐹 Fn 𝑋)
11		fvelrnb 6728	. . . . . . 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 234	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏)
14		simpllr 774	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg)
15	1	kqid 22338	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
16	15	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
17		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽))
18		cnima 21875	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑎) ∈ 𝐽)
19	16, 17, 18	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → (^◡𝐹 “ 𝑎) ∈ 𝐽)
20	9	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
21	20	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
22		elpreima 6830	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑧 ∈ (^◡𝐹 “ 𝑎))
25		regsep 21944	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Reg ∧ (^◡𝐹 “ 𝑎) ∈ 𝐽 ∧ 𝑧 ∈ (^◡𝐹 “ 𝑎)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)))
26	14, 19, 24, 25	syl3anc 1367	. . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 22344	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
30	27, 28, 29	syl2anc 586	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 22340	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑤 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
34	27, 28, 32, 33	syl3anc 1367	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 21523	. . . . . . . . . . . . . . . . . 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 4870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
39	38	ad2antrl 726	. . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	clscld 21657	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
42	37, 39, 41	syl2anc 586	. . . . . . . . . . . . . . . 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 22345	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
44	27, 42, 43	syl2anc 586	. . . . . . . . . . . . . . 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 21666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
46	37, 39, 45	syl2anc 586	. . . . . . . . . . . . . . . 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 5967	. . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . 16 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
50	49	clsss2 21682	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 6455	. . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 6439	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝐹 “ 𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
57	54, 55, 56	syl2anc 586	. . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . 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 2903	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((𝐹‘𝑧) ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)))
60		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹 “ 𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)))
61	60	sseq1d 4000	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹 “ 𝑤) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎))
62	59, 61	anbi12d 632	. . . . . . . . . . . . . 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 3625	. . . . . . . . . . . . 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 3293	. . . . . . . . . . 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 459	. . . . . . . . . 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 2902	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑎 ↔ 𝑏 ∈ 𝑎))
68		eleq1 2902	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑚 ↔ 𝑏 ∈ 𝑚))
69	68	anbi1d 631	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) = 𝑏 → (((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ (𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
70	69	rexbidv 3299	. . . . . . . . . . 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 347	. . . . . . . . . 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 247	. . . . . . . . 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 409	. . . . . . 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 650	. . . . . 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 3286	. . . . 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 469	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑎 ∈ (KQ‘𝐽) ∧ 𝑏 ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
79	78	ralrimivva 3193	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏 ∈ 𝑎 ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
80		isreg 21942	. 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 585	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∪ cuni 4840 ↦ cmpt 5148 ^◡ccnv 5556 ran crn 5558 “ cima 5560 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 Topctop 21503 TopOnctopon 21520 Clsdccld 21626 clsccl 21628 Cn ccn 21834 Regcreg 21919 KQckq 22303

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-qtop 16782 df-top 21504 df-topon 21521 df-cld 21629 df-cls 21631 df-cn 21837 df-reg 21926 df-kq 22304

This theorem is referenced by: kqreg 22361

W3C validator