Theorem kgeni 23584

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 4177	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽))
2		in32 4179	. . . . 5 ⊢ ((𝐴 ∩ 𝐾) ∩ ∪ 𝐽) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
3	1, 2	eqtr3i 2786	. . . 4 ⊢ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
4		df-kgen 23581	. . . . . . . . . . 11 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑦 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑦) ∈ Comp → (𝑥 ∩ 𝑦) ∈ (𝑗 ↾t 𝑦))})
5	4	mptrcl 6979	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
6	5	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
7		toptopon2 22965	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
8	6, 7	sylib 220	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘∪ 𝐽))
9		simpl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
10		elkgen 23583	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
11	10	biimpa 480	. . . . . . . 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 593	. . . . . . 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 498	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐴 ⊆ ∪ 𝐽)
14		dfss2 3920	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
15	13, 14	sylib 220	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ ∪ 𝐽) = 𝐴)
16	15	ineq1d 4169	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
17	3, 16	eqtrid 2808	. . 3 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = (𝐴 ∩ 𝐾))
18		cmptop 23442	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Comp → (𝐽 ↾t 𝐾) ∈ Top)
19	18	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Top)
20		restrcl 23204	. . . . . . . 8 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
21	20	simprd 499	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → 𝐾 ∈ V)
22	19, 21	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐾 ∈ V)
23		eqid 2761	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	restin 23213	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
25	6, 22, 24	syl2anc 593	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
26		simpr 488	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Comp)
27	25, 26	eqeltrrd 2862	. . . 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 2846	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑦) ∈ Comp ↔ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp))
30		ineq2 4164	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)))
31	30, 28	eleq12d 2855	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦) ↔ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽))))
32	29, 31	imbi12d 346	. . . . 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 499	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → (𝐴 ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
34		inss2 4187	. . . . . 6 ⊢ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
35		inex1g 5272	. . . . . . 7 ⊢ (𝐾 ∈ V → (𝐾 ∩ ∪ 𝐽) ∈ V)
36		elpwg 4555	. . . . . . 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 260	. . . . 5 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐾 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽)
39	32, 33, 38	rspcdva 3581	. . . 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 2862	. 2 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
42	41, 25	eleqtrrd 2864	1 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4552  ∪ cuni 4862  ‘cfv 6515  (class class class)co 7390   ↾_t crest 17439  Topctop 22940  TopOnctopon 22957  Compccmp 23433  𝑘Genckgen 23580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-rest 17441  df-top 22941  df-topon 22958  df-cmp 23434  df-kgen 23581

This theorem is referenced by:  kgentopon  23585  kgencmp  23592  kgenidm  23594  llycmpkgen2  23597  1stckgen  23601  kgencn3  23605  txkgen  23699