Theorem kgeni 23527

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 4163	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽))
2		in32 4165	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
3	1, 2	eqtr3i 2765	. . . 4 ⊢ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
4		df-kgen 23524	. . . . . . . . . . 11 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑦 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑦) ∈ Comp → (𝑥 ∩ 𝑦) ∈ (𝑗 ↾t 𝑦))})
5	4	mptrcl 6952	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
6	5	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
7		toptopon2 22908	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
8	6, 7	sylib 219	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘∪ 𝐽))
9		simpl 483	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
10		elkgen 23526	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
11	10	biimpa 477	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐴 ∈ (𝑘Gen‘𝐽)) → (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))))
12	8, 9, 11	syl2anc 590	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))))
13	12	simpld 495	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ⊆ ∪ 𝐽)
14		dfss2 3908	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
15	13, 14	sylib 219	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ ∪ 𝐽) = 𝐴)
16	15	ineq1d 4155	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
17	3, 16	eqtrid 2787	. . 3 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = (𝐴 ∩ 𝐾))
18		cmptop 23385	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Comp → (𝐽 ↾t 𝐾) ∈ Top)
19	18	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Top)
20		restrcl 23147	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
21	20	simprd 496	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → 𝐾 ∈ V)
22	19, 21	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐾 ∈ V)
23		eqid 2740	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	restin 23156	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
25	6, 22, 24	syl2anc 590	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
26		simpr 485	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Comp)
27	25, 26	eqeltrrd 2841	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp)
28		oveq2 7371	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑦) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
29	28	eleq1d 2825	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑦) ∈ Comp ↔ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp))
30		ineq2 4150	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)))
31	30, 28	eleq12d 2834	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦) ↔ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽))))
32	29, 31	imbi12d 345	. . . . 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)))))
33	12	simprd 496	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
34		inss2 4173	. . . . . 6 ⊢ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
35		inex1g 5254	. . . . . . 7 ⊢ (𝐾 ∈ V → (𝐾 ∩ ∪ 𝐽) ∈ V)
36		elpwg 4539	. . . . . . 7 ⊢ ((𝐾 ∩ ∪ 𝐽) ∈ V → ((𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽))
37	22, 35, 36	3syl 18	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽))
38	34, 37	mpbiri 259	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽)
39	32, 33, 38	rspcdva 3568	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽))))
40	27, 39	mpd 15	. . 3 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
41	17, 40	eqeltrrd 2841	. 2 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
42	41, 25	eleqtrrd 2843	1 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845  ‘cfv 6492  (class class class)co 7363   ↾_t crest 17381  Topctop 22883  TopOnctopon 22900  Compccmp 23376  𝑘Genckgen 23523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-rest 17383  df-top 22884  df-topon 22901  df-cmp 23377  df-kgen 23524

This theorem is referenced by:  kgentopon  23528  kgencmp  23535  kgenidm  23537  llycmpkgen2  23540  1stckgen  23544  kgencn3  23548  txkgen  23642