Theorem kgeni 23431

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 4194	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽))
2		in32 4196	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
3	1, 2	eqtr3i 2755	. . . 4 ⊢ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
4		df-kgen 23428	. . . . . . . . . . 11 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑦 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑦) ∈ Comp → (𝑥 ∩ 𝑦) ∈ (𝑗 ↾t 𝑦))})
5	4	mptrcl 6980	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
7		toptopon2 22812	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
8	6, 7	sylib 218	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘∪ 𝐽))
9		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
10		elkgen 23430	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
11	10	biimpa 476	. . . . . . . 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 584	. . . . . . 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 494	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ⊆ ∪ 𝐽)
14		dfss2 3935	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
15	13, 14	sylib 218	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ ∪ 𝐽) = 𝐴)
16	15	ineq1d 4185	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
17	3, 16	eqtrid 2777	. . 3 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = (𝐴 ∩ 𝐾))
18		cmptop 23289	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Comp → (𝐽 ↾t 𝐾) ∈ Top)
19	18	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Top)
20		restrcl 23051	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
21	20	simprd 495	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → 𝐾 ∈ V)
22	19, 21	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐾 ∈ V)
23		eqid 2730	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	restin 23060	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
25	6, 22, 24	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
26		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Comp)
27	25, 26	eqeltrrd 2830	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp)
28		oveq2 7398	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑦) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
29	28	eleq1d 2814	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑦) ∈ Comp ↔ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp))
30		ineq2 4180	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)))
31	30, 28	eleq12d 2823	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦) ↔ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽))))
32	29, 31	imbi12d 344	. . . . 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 495	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
34		inss2 4204	. . . . . 6 ⊢ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
35		inex1g 5277	. . . . . . 7 ⊢ (𝐾 ∈ V → (𝐾 ∩ ∪ 𝐽) ∈ V)
36		elpwg 4569	. . . . . . 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 258	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽)
39	32, 33, 38	rspcdva 3592	. . . 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 2830	. 2 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
42	41, 25	eleqtrrd 2832	1 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ‘cfv 6514  (class class class)co 7390   ↾_t crest 17390  Topctop 22787  TopOnctopon 22804  Compccmp 23280  𝑘Genckgen 23427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-rest 17392  df-top 22788  df-topon 22805  df-cmp 23281  df-kgen 23428

This theorem is referenced by:  kgentopon  23432  kgencmp  23439  kgenidm  23441  llycmpkgen2  23444  1stckgen  23448  kgencn3  23452  txkgen  23546