Theorem rncmp 19764

Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
rncmp	|- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( K ↾t ran F ) e. Comp )

Proof of Theorem rncmp

Step	Hyp	Ref	Expression
1		simpl 457	. 2 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> J e. Comp )
2		eqid 2467	. . . . . . 7 |- U. J = U. J
3		eqid 2467	. . . . . . 7 |- U. K = U. K
4	2, 3	cnf 19615	. . . . . 6 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
5	4	adantl 466	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F : U. J --> U. K )
6		ffn 5737	. . . . 5 |- ( F : U. J --> U. K -> F Fn U. J )
7	5, 6	syl 16	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F Fn U. J )
8		dffn4 5807	. . . 4 |- ( F Fn U. J <-> F : U. J -onto-> ran F )
9	7, 8	sylib 196	. . 3 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F : U. J -onto-> ran F )
10		cntop2 19610	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
11	10	adantl 466	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> K e. Top )
12		frn 5743	. . . . . 6 |- ( F : U. J --> U. K -> ran F C_ U. K )
13	5, 12	syl 16	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ran F C_ U. K )
14	3	restuni 19531	. . . . 5 |- ( ( K e. Top /\ ran F C_ U. K ) -> ran F = U. ( K ↾t ran F ) )
15	11, 13, 14	syl2anc 661	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ran F = U. ( K ↾t ran F ) )
16		foeq3 5799	. . . 4 |- ( ran F = U. ( K ↾t ran F ) -> ( F : U. J -onto-> ran F <-> F : U. J -onto-> U. ( K ↾t ran F ) ) )
17	15, 16	syl 16	. . 3 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( F : U. J -onto-> ran F <-> F : U. J -onto-> U. ( K ↾t ran F ) ) )
18	9, 17	mpbid 210	. 2 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F : U. J -onto-> U. ( K ↾t ran F ) )
19		simpr 461	. . 3 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F e. ( J Cn K ) )
20	3	toptopon 19303	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
21	11, 20	sylib 196	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> K e. (TopOn ` U. K ) )
22		ssid 3528	. . . . 5 |- ran F C_ ran F
23	22	a1i 11	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ran F C_ ran F )
24		cnrest2 19655	. . . 4 |- ( ( K e. (TopOn ` U. K ) /\ ran F C_ ran F /\ ran F C_ U. K ) -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K ↾t ran F ) ) ) )
25	21, 23, 13, 24	syl3anc 1228	. . 3 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K ↾t ran F ) ) ) )
26	19, 25	mpbid 210	. 2 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F e. ( J Cn ( K ↾t ran F ) ) )
27		eqid 2467	. . 3 |- U. ( K ↾t ran F ) = U. ( K ↾t ran F )
28	27	cncmp 19760	. 2 |- ( ( J e. Comp /\ F : U. J -onto-> U. ( K ↾t ran F ) /\ F e. ( J Cn ( K ↾t ran F ) ) ) -> ( K ↾t ran F ) e. Comp )
29	1, 18, 26, 28	syl3anc 1228	1 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( K ↾t ran F ) e. Comp )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   C_ wss 3481   U.cuni 4251   ran crn 5006   Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295   ↾_t crest 14693   Topctop 19263  TopOnctopon 19264   Cn ccn 19593   Compccmp 19754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-fin 7532  df-fi 7883  df-rest 14695  df-topgen 14716  df-top 19268  df-bases 19270  df-topon 19271  df-cn 19596  df-cmp 19755

This theorem is referenced by:  imacmp  19765  kgencn2  19926  bndth  21326