Theorem kgencmp 17608

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 17603	. . . 4 |- ( J e. Top -> (𝑘Gen ` J ) e. Top )
2	1	adantr 453	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> (𝑘Gen ` J ) e. Top )
3		kgenss 17606	. . . 4 |- ( J e. Top -> J C_ (𝑘Gen ` J ) )
4	3	adantr 453	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> J C_ (𝑘Gen ` J ) )
5		ssrest 17271	. . 3 |- ( ( (𝑘Gen ` J ) e. Top /\ J C_ (𝑘Gen ` J ) ) -> ( J ↾t K ) C_ ( (𝑘Gen ` J ) ↾t K ) )
6	2, 4, 5	syl2anc 644	. 2 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) C_ ( (𝑘Gen ` J ) ↾t K ) )
7		cmptop 17489	. . . . . 6 |- ( ( J ↾t K ) e. Comp -> ( J ↾t K ) e. Top )
8	7	adantl 454	. . . . 5 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) e. Top )
9		restrcl 17252	. . . . . 6 |- ( ( J ↾t K ) e. Top -> ( J e. _V /\ K e. _V ) )
10	9	simprd 451	. . . . 5 |- ( ( J ↾t K ) e. Top -> K e. _V )
11	8, 10	syl 16	. . . 4 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> K e. _V )
12		restval 13685	. . . 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 644	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( (𝑘Gen ` J ) ↾t K ) = ran ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) )
14		simpr 449	. . . . . 6 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> x e. (𝑘Gen ` J ) )
15		simplr 733	. . . . . 6 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> ( J ↾t K ) e. Comp )
16		kgeni 17600	. . . . . 6 |- ( ( x e. (𝑘Gen ` J ) /\ ( J ↾t K ) e. Comp ) -> ( x i^i K ) e. ( J ↾t K ) )
17	14, 15, 16	syl2anc 644	. . . . 5 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> ( x i^i K ) e. ( J ↾t K ) )
18		eqid 2442	. . . . 5 |- ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) = ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) )
19	17, 18	fmptd 5922	. . . 4 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) : (𝑘Gen ` J ) --> ( J ↾t K ) )
20		frn 5626	. . . 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 16	. . 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 3368	. 2 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( (𝑘Gen ` J ) ↾t K ) C_ ( J ↾t K ) )
23	6, 22	eqssd 3351	1 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) = ( (𝑘Gen ` J ) ↾t K ) )

Syntax hints:   -> wi 4   /\ wa 360   = wceq 1653   e. wcel 1727   _Vcvv 2962   i^i cin 3305   C_ wss 3306   e. cmpt 4291   ran crn 4908   -->wf 5479   ` cfv 5483  (class class class)co 6110   ↾_t crest 13679   Topctop 16989   Compccmp 17480  𝑘Genckgen 17596

This theorem is referenced by:  kgencmp2  17609  kgenidm  17610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-recs 6662  df-rdg 6697  df-oadd 6757  df-er 6934  df-en 7139  df-fin 7142  df-fi 7445  df-rest 13681  df-topgen 13698  df-top 16994  df-bases 16996  df-topon 16997  df-cmp 17481  df-kgen 17597