Theorem kgen2ss 22804

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 1135	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2		elpwi 4553	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
3		resttopon 22410	. . . . . . . . 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 1136	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6		resttopon 22410	. . . . . . . . . . 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 22161	. . . . . . . . . 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 6823	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘∪ (𝐾 ↾t 𝑘)))
11	4, 10	eleqtrd 2839	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)))
12		simpl2 1191	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13		topontop 22160	. . . . . . . . 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 1192	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽 ⊆ 𝐾)
16		ssrest 22425	. . . . . . . 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 2736	. . . . . . . . . 10 ⊢ ∪ (𝐾 ↾t 𝑘) = ∪ (𝐾 ↾t 𝑘)
19	18	sscmp 22654	. . . . . . . . 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 1125	. . . . . . . 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 1120	. . . . . . 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 3930	. . . . . 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 3160	. . . 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 22785	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
28	27	3ad2ant1 1132	. . 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 22785	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
30	29	3ad2ant2 1133	. . 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 293	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
32	31	ssrdv 3937	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4546  ∪ cuni 4851  ‘cfv 6473  (class class class)co 7329   ↾_t crest 17220  Topctop 22140  TopOnctopon 22157  Compccmp 22635  𝑘Genckgen 22782

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-en 8797  df-fin 8800  df-fi 9260  df-rest 17222  df-topgen 17243  df-top 22141  df-topon 22158  df-bases 22194  df-cmp 22636  df-kgen 22783

This theorem is referenced by: (None)