Theorem rncmp 19141

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 2454	. . . . . . 7 |- U. J = U. J
3		eqid 2454	. . . . . . 7 |- U. K = U. K
4	2, 3	cnf 18992	. . . . . 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 5670	. . . . 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 5737	. . . 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 18987	. . . . . 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 5676	. . . . . 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 18908	. . . . 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 5729	. . . 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 18680	. . . . 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 3486	. . . . 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 19032	. . . 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 1219	. . 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 2454	. . 3 |- U. ( K ↾t ran F ) = U. ( K ↾t ran F )
28	27	cncmp 19137	. 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 1219	1 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( K ↾t ran F ) e. Comp )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   C_ wss 3439   U.cuni 4202   ran crn 4952   Fn wfn 5524   -->wf 5525   -onto->wfo 5527   ` cfv 5529  (class class class)co 6203   ↾_t crest 14482   Topctop 18640  TopOnctopon 18641   Cn ccn 18970   Compccmp 19131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-fin 7427  df-fi 7776  df-rest 14484  df-topgen 14505  df-top 18645  df-bases 18647  df-topon 18648  df-cn 18973  df-cmp 19132

This theorem is referenced by:  imacmp  19142  kgencn2  19272  bndth  20672