Theorem rncmp 18899

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 454	. 2 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> J e. Comp )
2		eqid 2441	. . . . . . 7 |- U. J = U. J
3		eqid 2441	. . . . . . 7 |- U. K = U. K
4	2, 3	cnf 18750	. . . . . 6 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
5	4	adantl 463	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F : U. J --> U. K )
6		ffn 5556	. . . . 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 5623	. . . 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 18745	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
11	10	adantl 463	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> K e. Top )
12		frn 5562	. . . . . 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 18666	. . . . 5 |- ( ( K e. Top /\ ran F C_ U. K ) -> ran F = U. ( K ↾t ran F ) )
15	11, 13, 14	syl2anc 656	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ran F = U. ( K ↾t ran F ) )
16		foeq3 5615	. . . 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 458	. . 3 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F e. ( J Cn K ) )
20	3	toptopon 18438	. . . . 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 3372	. . . . 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 18790	. . . 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 1213	. . 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 2441	. . 3 |- U. ( K ↾t ran F ) = U. ( K ↾t ran F )
28	27	cncmp 18895	. 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 1213	1 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( K ↾t ran F ) e. Comp )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   C_ wss 3325   U.cuni 4088   ran crn 4837   Fn wfn 5410   -->wf 5411   -onto->wfo 5413   ` cfv 5415  (class class class)co 6090   ↾_t crest 14355   Topctop 18398  TopOnctopon 18399   Cn ccn 18728   Compccmp 18889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-fin 7310  df-fi 7657  df-rest 14357  df-topgen 14378  df-top 18403  df-bases 18405  df-topon 18406  df-cn 18731  df-cmp 18890

This theorem is referenced by:  imacmp  18900  kgencn2  19030  bndth  20430