Theorem rncmp 20488

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 464	. 2 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> J e. Comp )
2		eqid 2471	. . . . . . 7 |- U. J = U. J
3		eqid 2471	. . . . . . 7 |- U. K = U. K
4	2, 3	cnf 20339	. . . . . 6 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
5	4	adantl 473	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F : U. J --> U. K )
6		ffn 5739	. . . . 5 |- ( F : U. J --> U. K -> F Fn U. J )
7	5, 6	syl 17	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F Fn U. J )
8		dffn4 5812	. . . 4 |- ( F Fn U. J <-> F : U. J -onto-> ran F )
9	7, 8	sylib 201	. . 3 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F : U. J -onto-> ran F )
10		cntop2 20334	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
11	10	adantl 473	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> K e. Top )
12		frn 5747	. . . . . 6 |- ( F : U. J --> U. K -> ran F C_ U. K )
13	5, 12	syl 17	. . . . 5 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ran F C_ U. K )
14	3	restuni 20255	. . . . 5 |- ( ( K e. Top /\ ran F C_ U. K ) -> ran F = U. ( K ↾t ran F ) )
15	11, 13, 14	syl2anc 673	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ran F = U. ( K ↾t ran F ) )
16		foeq3 5804	. . . 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 17	. . 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 215	. 2 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F : U. J -onto-> U. ( K ↾t ran F ) )
19		simpr 468	. . 3 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F e. ( J Cn K ) )
20	3	toptopon 20025	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
21	11, 20	sylib 201	. . . 4 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> K e. (TopOn ` U. K ) )
22		ssid 3437	. . . . 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 20379	. . . 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 1292	. . 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 215	. 2 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> F e. ( J Cn ( K ↾t ran F ) ) )
27		eqid 2471	. . 3 |- U. ( K ↾t ran F ) = U. ( K ↾t ran F )
28	27	cncmp 20484	. 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 1292	1 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( K ↾t ran F ) e. Comp )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   C_ wss 3390   U.cuni 4190   ran crn 4840   Fn wfn 5584   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   ↾_t crest 15397   Topctop 19994  TopOnctopon 19995   Cn ccn 20317   Compccmp 20478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-cmp 20479

This theorem is referenced by:  imacmp  20489  kgencn2  20649  bndth  22064