Theorem kgen2ss 23465

Description: The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
kgen2ss	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Proof of Theorem kgen2ss

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2		elpwi 4552	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
3		resttopon 23071	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
4	1, 2, 3	syl2an 596	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
5		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6		resttopon 23071	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
7	5, 2, 6	syl2an 596	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
8		toponuni 22824	. . . . . . . . . 10 ⊢ ((𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 = ∪ (𝐾 ↾t 𝑘))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 = ∪ (𝐾 ↾t 𝑘))
10	9	fveq2d 6821	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘∪ (𝐾 ↾t 𝑘)))
11	4, 10	eleqtrd 2833	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)))
12		simpl2 1193	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13		topontop 22823	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15		simpl3 1194	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽 ⊆ 𝐾)
16		ssrest 23086	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
17	14, 15, 16	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
18		eqid 2731	. . . . . . . . . 10 ⊢ ∪ (𝐾 ↾t 𝑘) = ∪ (𝐾 ↾t 𝑘)
19	18	sscmp 23315	. . . . . . . . 9 ⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)) ∧ (𝐾 ↾t 𝑘) ∈ Comp ∧ (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘)) → (𝐽 ↾t 𝑘) ∈ Comp)
20	19	3com23 1126	. . . . . . . 8 ⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)) ∧ (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘) ∧ (𝐾 ↾t 𝑘) ∈ Comp) → (𝐽 ↾t 𝑘) ∈ Comp)
21	20	3expia 1121	. . . . . . 7 ⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)) ∧ (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘)) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Comp))
22	11, 17, 21	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Comp))
23	17	sseld 3928	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))
24	22, 23	imim12d 81	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘))))
25	24	ralimdva 3144	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘))))
26	25	anim2d 612	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → ((𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) → (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
27		elkgen 23446	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
28	27	3ad2ant1 1133	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
29		elkgen 23446	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
30	29	3ad2ant2 1134	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
31	26, 28, 30	3imtr4d 294	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
32	31	ssrdv 3935	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4545  ∪ cuni 4854  ‘cfv 6476  (class class class)co 7341   ↾_t crest 17319  Topctop 22803  TopOnctopon 22820  Compccmp 23296  𝑘Genckgen 23443

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-en 8865  df-fin 8868  df-fi 9290  df-rest 17321  df-topgen 17342  df-top 22804  df-topon 22821  df-bases 22856  df-cmp 23297  df-kgen 23444

This theorem is referenced by: (None)