Theorem kgencmp 21348

Description: The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
kgencmp	|- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) = ( (𝑘Gen ` J ) ↾t K ) )

Proof of Theorem kgencmp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kgenftop 21343	. . . 4 |- ( J e. Top -> (𝑘Gen ` J ) e. Top )
2	1	adantr 481	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> (𝑘Gen ` J ) e. Top )
3		kgenss 21346	. . . 4 |- ( J e. Top -> J C_ (𝑘Gen ` J ) )
4	3	adantr 481	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> J C_ (𝑘Gen ` J ) )
5		ssrest 20980	. . 3 |- ( ( (𝑘Gen ` J ) e. Top /\ J C_ (𝑘Gen ` J ) ) -> ( J ↾t K ) C_ ( (𝑘Gen ` J ) ↾t K ) )
6	2, 4, 5	syl2anc 693	. 2 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) C_ ( (𝑘Gen ` J ) ↾t K ) )
7		cmptop 21198	. . . . . 6 |- ( ( J ↾t K ) e. Comp -> ( J ↾t K ) e. Top )
8	7	adantl 482	. . . . 5 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) e. Top )
9		restrcl 20961	. . . . . 6 |- ( ( J ↾t K ) e. Top -> ( J e. _V /\ K e. _V ) )
10	9	simprd 479	. . . . 5 |- ( ( J ↾t K ) e. Top -> K e. _V )
11	8, 10	syl 17	. . . 4 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> K e. _V )
12		restval 16087	. . . 4 |- ( ( (𝑘Gen ` J ) e. Top /\ K e. _V ) -> ( (𝑘Gen ` J ) ↾t K ) = ran ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) )
13	2, 11, 12	syl2anc 693	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( (𝑘Gen ` J ) ↾t K ) = ran ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) )
14		simpr 477	. . . . . 6 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> x e. (𝑘Gen ` J ) )
15		simplr 792	. . . . . 6 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> ( J ↾t K ) e. Comp )
16		kgeni 21340	. . . . . 6 |- ( ( x e. (𝑘Gen ` J ) /\ ( J ↾t K ) e. Comp ) -> ( x i^i K ) e. ( J ↾t K ) )
17	14, 15, 16	syl2anc 693	. . . . 5 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> ( x i^i K ) e. ( J ↾t K ) )
18		eqid 2622	. . . . 5 |- ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) = ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) )
19	17, 18	fmptd 6385	. . . 4 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) : (𝑘Gen ` J ) --> ( J ↾t K ) )
20		frn 6053	. . . 4 |- ( ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) : (𝑘Gen ` J ) --> ( J ↾t K ) -> ran ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) C_ ( J ↾t K ) )
21	19, 20	syl 17	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ran ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) C_ ( J ↾t K ) )
22	13, 21	eqsstrd 3639	. 2 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( (𝑘Gen ` J ) ↾t K ) C_ ( J ↾t K ) )
23	6, 22	eqssd 3620	1 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) = ( (𝑘Gen ` J ) ↾t K ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698   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:  kgencmp2  21349  kgenidm  21350