Theorem kgeni 21150

Description: Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
kgeni	⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾))

Proof of Theorem kgeni

Dummy variables 𝑦 𝑥 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inass 3785	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽))
2		in32 3787	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
3	1, 2	eqtr3i 2634	. . . 4 ⊢ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
4		df-kgen 21147	. . . . . . . . . . . 12 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑦 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑦) ∈ Comp → (𝑥 ∩ 𝑦) ∈ (𝑗 ↾t 𝑦))})
5	4	dmmptss 5548	. . . . . . . . . . 11 ⊢ dom 𝑘Gen ⊆ Top
6		elfvdm 6130	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ dom 𝑘Gen)
7	5, 6	sseldi 3566	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
9		eqid 2610	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	toptopon 20548	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
11	8, 10	sylib 207	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘∪ 𝐽))
12		simpl 472	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
13		elkgen 21149	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
14	13	biimpa 500	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐴 ∈ (𝑘Gen‘𝐽)) → (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))))
15	11, 12, 14	syl2anc 691	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))))
16	15	simpld 474	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ⊆ ∪ 𝐽)
17		df-ss 3554	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
18	16, 17	sylib 207	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ ∪ 𝐽) = 𝐴)
19	18	ineq1d 3775	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
20	3, 19	syl5eq 2656	. . 3 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = (𝐴 ∩ 𝐾))
21		inss2 3796	. . . . 5 ⊢ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
22		cmptop 21008	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Comp → (𝐽 ↾t 𝐾) ∈ Top)
23	22	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Top)
24		restrcl 20771	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
25	24	simprd 478	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → 𝐾 ∈ V)
26	23, 25	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐾 ∈ V)
27		inex1g 4729	. . . . . 6 ⊢ (𝐾 ∈ V → (𝐾 ∩ ∪ 𝐽) ∈ V)
28		elpwg 4116	. . . . . 6 ⊢ ((𝐾 ∩ ∪ 𝐽) ∈ V → ((𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽))
29	26, 27, 28	3syl 18	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽))
30	21, 29	mpbiri 247	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽)
31	15	simprd 478	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
32	9	restin 20780	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
33	8, 26, 32	syl2anc 691	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
34		simpr 476	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Comp)
35	33, 34	eqeltrrd 2689	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp)
36		oveq2 6557	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑦) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
37	36	eleq1d 2672	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑦) ∈ Comp ↔ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp))
38		ineq2 3770	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)))
39	38, 36	eleq12d 2682	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦) ↔ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽))))
40	37, 39	imbi12d 333	. . . . 5 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)) ↔ ((𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))))
41	40	rspcv 3278	. . . 4 ⊢ ((𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 → (∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)) → ((𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))))
42	30, 31, 35, 41	syl3c 64	. . 3 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
43	20, 42	eqeltrrd 2689	. 2 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
44	43, 33	eleqtrrd 2691	1 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  Topctop 20517  TopOnctopon 20518  Compccmp 20999  𝑘Genckgen 21146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906  df-top 20521  df-topon 20523  df-cmp 21000  df-kgen 21147

This theorem is referenced by:  kgentopon  21151  kgencmp  21158  kgenidm  21160  llycmpkgen2  21163  1stckgen  21167  kgencn3  21171  txkgen  21265