Theorem kgencmp 17538

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 17533	. . . 4 |- ( J e. Top -> (𝑘Gen ` J ) e. Top )
2	1	adantr 452	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> (𝑘Gen ` J ) e. Top )
3		kgenss 17536	. . . 4 |- ( J e. Top -> J C_ (𝑘Gen ` J ) )
4	3	adantr 452	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> J C_ (𝑘Gen ` J ) )
5		ssrest 17202	. . 3 |- ( ( (𝑘Gen ` J ) e. Top /\ J C_ (𝑘Gen ` J ) ) -> ( J ↾t K ) C_ ( (𝑘Gen ` J ) ↾t K ) )
6	2, 4, 5	syl2anc 643	. 2 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) C_ ( (𝑘Gen ` J ) ↾t K ) )
7		cmptop 17420	. . . . . 6 |- ( ( J ↾t K ) e. Comp -> ( J ↾t K ) e. Top )
8	7	adantl 453	. . . . 5 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) e. Top )
9		restrcl 17183	. . . . . 6 |- ( ( J ↾t K ) e. Top -> ( J e. _V /\ K e. _V ) )
10	9	simprd 450	. . . . 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 13617	. . . 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 643	. . 3 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( (𝑘Gen ` J ) ↾t K ) = ran ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) )
14		simpr 448	. . . . . 6 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> x e. (𝑘Gen ` J ) )
15		simplr 732	. . . . . 6 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> ( J ↾t K ) e. Comp )
16		kgeni 17530	. . . . . 6 |- ( ( x e. (𝑘Gen ` J ) /\ ( J ↾t K ) e. Comp ) -> ( x i^i K ) e. ( J ↾t K ) )
17	14, 15, 16	syl2anc 643	. . . . 5 |- ( ( ( J e. Top /\ ( J ↾t K ) e. Comp ) /\ x e. (𝑘Gen ` J ) ) -> ( x i^i K ) e. ( J ↾t K ) )
18		eqid 2412	. . . . 5 |- ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) = ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) )
19	17, 18	fmptd 5860	. . . 4 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( x e. (𝑘Gen ` J ) |-> ( x i^i K ) ) : (𝑘Gen ` J ) --> ( J ↾t K ) )
20		frn 5564	. . . 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 3350	. 2 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( (𝑘Gen ` J ) ↾t K ) C_ ( J ↾t K ) )
23	6, 22	eqssd 3333	1 |- ( ( J e. Top /\ ( J ↾t K ) e. Comp ) -> ( J ↾t K ) = ( (𝑘Gen ` J ) ↾t K ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2924   i^i cin 3287   C_ wss 3288   e. cmpt 4234   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   ↾_t crest 13611   Topctop 16921   Compccmp 17411  𝑘Genckgen 17526

This theorem is referenced by:  kgencmp2  17539  kgenidm  17540

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-oadd 6695  df-er 6872  df-en 7077  df-fin 7080  df-fi 7382  df-rest 13613  df-topgen 13630  df-top 16926  df-bases 16928  df-topon 16929  df-cmp 17412  df-kgen 17527