Theorem kgen2ss 21358

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

Assertion

Ref	Expression
kgen2ss	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> (𝑘Gen ` J ) C_ (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 1061	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> J e. (TopOn ` X ) )
2		elpwi 4168	. . . . . . . . 9 |- ( k e. ~P X -> k C_ X )
3		resttopon 20965	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ k C_ X ) -> ( J ↾t k ) e. (TopOn ` k ) )
4	1, 2, 3	syl2an 494	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> ( J ↾t k ) e. (TopOn ` k ) )
5		simp2 1062	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> K e. (TopOn ` X ) )
6		resttopon 20965	. . . . . . . . . . 11 |- ( ( K e. (TopOn ` X ) /\ k C_ X ) -> ( K ↾t k ) e. (TopOn ` k ) )
7	5, 2, 6	syl2an 494	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> ( K ↾t k ) e. (TopOn ` k ) )
8		toponuni 20719	. . . . . . . . . 10 |- ( ( K ↾t k ) e. (TopOn ` k ) -> k = U. ( K ↾t k ) )
9	7, 8	syl 17	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> k = U. ( K ↾t k ) )
10	9	fveq2d 6195	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> (TopOn ` k ) = (TopOn ` U. ( K ↾t k ) ) )
11	4, 10	eleqtrd 2703	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> ( J ↾t k ) e. (TopOn ` U. ( K ↾t k ) ) )
12		simpl2 1065	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> K e. (TopOn ` X ) )
13		topontop 20718	. . . . . . . . 9 |- ( K e. (TopOn ` X ) -> K e. Top )
14	12, 13	syl 17	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> K e. Top )
15		simpl3 1066	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> J C_ K )
16		ssrest 20980	. . . . . . . 8 |- ( ( K e. Top /\ J C_ K ) -> ( J ↾t k ) C_ ( K ↾t k ) )
17	14, 15, 16	syl2anc 693	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> ( J ↾t k ) C_ ( K ↾t k ) )
18		eqid 2622	. . . . . . . . . 10 |- U. ( K ↾t k ) = U. ( K ↾t k )
19	18	sscmp 21208	. . . . . . . . 9 |- ( ( ( J ↾t k ) e. (TopOn ` U. ( K ↾t k ) ) /\ ( K ↾t k ) e. Comp /\ ( J ↾t k ) C_ ( K ↾t k ) ) -> ( J ↾t k ) e. Comp )
20	19	3com23 1271	. . . . . . . 8 |- ( ( ( J ↾t k ) e. (TopOn ` U. ( K ↾t k ) ) /\ ( J ↾t k ) C_ ( K ↾t k ) /\ ( K ↾t k ) e. Comp ) -> ( J ↾t k ) e. Comp )
21	20	3expia 1267	. . . . . . 7 |- ( ( ( J ↾t k ) e. (TopOn ` U. ( K ↾t k ) ) /\ ( J ↾t k ) C_ ( K ↾t k ) ) -> ( ( K ↾t k ) e. Comp -> ( J ↾t k ) e. Comp ) )
22	11, 17, 21	syl2anc 693	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> ( ( K ↾t k ) e. Comp -> ( J ↾t k ) e. Comp ) )
23	17	sseld 3602	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> ( ( x i^i k ) e. ( J ↾t k ) -> ( x i^i k ) e. ( K ↾t k ) ) )
24	22, 23	imim12d 81	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) /\ k e. ~P X ) -> ( ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> ( ( K ↾t k ) e. Comp -> ( x i^i k ) e. ( K ↾t k ) ) ) )
25	24	ralimdva 2962	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> A. k e. ~P X ( ( K ↾t k ) e. Comp -> ( x i^i k ) e. ( K ↾t k ) ) ) )
26	25	anim2d 589	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ( x C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) -> ( x C_ X /\ A. k e. ~P X ( ( K ↾t k ) e. Comp -> ( x i^i k ) e. ( K ↾t k ) ) ) ) )
27		elkgen 21339	. . . 4 |- ( J e. (TopOn ` X ) -> ( x e. (𝑘Gen ` J ) <-> ( x C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) ) )
28	27	3ad2ant1 1082	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( x e. (𝑘Gen ` J ) <-> ( x C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) ) )
29		elkgen 21339	. . . 4 |- ( K e. (TopOn ` X ) -> ( x e. (𝑘Gen ` K ) <-> ( x C_ X /\ A. k e. ~P X ( ( K ↾t k ) e. Comp -> ( x i^i k ) e. ( K ↾t k ) ) ) ) )
30	29	3ad2ant2 1083	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( x e. (𝑘Gen ` K ) <-> ( x C_ X /\ A. k e. ~P X ( ( K ↾t k ) e. Comp -> ( x i^i k ) e. ( K ↾t k ) ) ) ) )
31	26, 28, 30	3imtr4d 283	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( x e. (𝑘Gen ` J ) -> x e. (𝑘Gen ` K ) ) )
32	31	ssrdv 3609	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> (𝑘Gen ` J ) C_ (𝑘Gen ` K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   Compccmp 21189  𝑘Genckgen 21336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-kgen 21337

This theorem is referenced by: (None)