Theorem kgeni 23455

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 2758	. . . 4 ⊢ (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾)
4		df-kgen 23452	. . . . . . . . . . 11 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑦 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑦) ∈ Comp → (𝑥 ∩ 𝑦) ∈ (𝑗 ↾t 𝑦))})
5	4	mptrcl 6946	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
7		toptopon2 22836	. . . . . . . . 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 23454	. . . . . . . . 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 3916	. . . . . 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 4168	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝐴 ∩ ∪ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
17	3, 16	eqtrid 2780	. . 3 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)) = (𝐴 ∩ 𝐾))
18		cmptop 23313	. . . . . . . 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 23075	. . . . . . . 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 2733	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	restin 23084	. . . . . 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 2834	. . . 4 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp)
28		oveq2 7362	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑦) = (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
29	28	eleq1d 2818	. . . . . 6 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑦) ∈ Comp ↔ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)) ∈ Comp))
30		ineq2 4163	. . . . . . 7 ⊢ (𝑦 = (𝐾 ∩ ∪ 𝐽) → (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ ∪ 𝐽)))
31	30, 28	eleq12d 2827	. . . . . 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 4187	. . . . . 6 ⊢ (𝐾 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
35		inex1g 5261	. . . . . . 7 ⊢ (𝐾 ∈ V → (𝐾 ∩ ∪ 𝐽) ∈ V)
36		elpwg 4554	. . . . . . 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 3574	. . . 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 2834	. 2 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t (𝐾 ∩ ∪ 𝐽)))
42	41, 25	eleqtrrd 2836	1 ⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4551  ∪ cuni 4860  ‘cfv 6488  (class class class)co 7354   ↾_t crest 17328  Topctop 22811  TopOnctopon 22828  Compccmp 23304  𝑘Genckgen 23451

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-rest 17330  df-top 22812  df-topon 22829  df-cmp 23305  df-kgen 23452

This theorem is referenced by:  kgentopon  23456  kgencmp  23463  kgenidm  23465  llycmpkgen2  23468  1stckgen  23472  kgencn3  23476  txkgen  23570