Theorem imacmp 21200

Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
imacmp	|- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( K ↾t ( F " A ) ) e. Comp )

Proof of Theorem imacmp

Step	Hyp	Ref	Expression
1		df-ima 5127	. . 3 |- ( F " A ) = ran ( F |` A )
2	1	oveq2i 6661	. 2 |- ( K ↾t ( F " A ) ) = ( K ↾t ran ( F |` A ) )
3		simpr 477	. . 3 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( J ↾t A ) e. Comp )
4		simpl 473	. . . . 5 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> F e. ( J Cn K ) )
5		inss2 3834	. . . . 5 |- ( A i^i U. J ) C_ U. J
6		eqid 2622	. . . . . 6 |- U. J = U. J
7	6	cnrest 21089	. . . . 5 |- ( ( F e. ( J Cn K ) /\ ( A i^i U. J ) C_ U. J ) -> ( F |` ( A i^i U. J ) ) e. ( ( J ↾t ( A i^i U. J ) ) Cn K ) )
8	4, 5, 7	sylancl 694	. . . 4 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( F |` ( A i^i U. J ) ) e. ( ( J ↾t ( A i^i U. J ) ) Cn K ) )
9		resdmres 5625	. . . . 5 |- ( F |` dom ( F |` A ) ) = ( F |` A )
10		dmres 5419	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
11		eqid 2622	. . . . . . . . . 10 |- U. K = U. K
12	6, 11	cnf 21050	. . . . . . . . 9 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
13		fdm 6051	. . . . . . . . 9 |- ( F : U. J --> U. K -> dom F = U. J )
14	4, 12, 13	3syl 18	. . . . . . . 8 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> dom F = U. J )
15	14	ineq2d 3814	. . . . . . 7 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( A i^i dom F ) = ( A i^i U. J ) )
16	10, 15	syl5eq 2668	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> dom ( F |` A ) = ( A i^i U. J ) )
17	16	reseq2d 5396	. . . . 5 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( F |` dom ( F |` A ) ) = ( F |` ( A i^i U. J ) ) )
18	9, 17	syl5eqr 2670	. . . 4 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( F |` A ) = ( F |` ( A i^i U. J ) ) )
19		cmptop 21198	. . . . . . 7 |- ( ( J ↾t A ) e. Comp -> ( J ↾t A ) e. Top )
20	19	adantl 482	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( J ↾t A ) e. Top )
21		restrcl 20961	. . . . . 6 |- ( ( J ↾t A ) e. Top -> ( J e. _V /\ A e. _V ) )
22	6	restin 20970	. . . . . 6 |- ( ( J e. _V /\ A e. _V ) -> ( J ↾t A ) = ( J ↾t ( A i^i U. J ) ) )
23	20, 21, 22	3syl 18	. . . . 5 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( J ↾t A ) = ( J ↾t ( A i^i U. J ) ) )
24	23	oveq1d 6665	. . . 4 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( ( J ↾t A ) Cn K ) = ( ( J ↾t ( A i^i U. J ) ) Cn K ) )
25	8, 18, 24	3eltr4d 2716	. . 3 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
26		rncmp 21199	. . 3 |- ( ( ( J ↾t A ) e. Comp /\ ( F |` A ) e. ( ( J ↾t A ) Cn K ) ) -> ( K ↾t ran ( F |` A ) ) e. Comp )
27	3, 25, 26	syl2anc 693	. 2 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( K ↾t ran ( F |` A ) ) e. Comp )
28	2, 27	syl5eqel 2705	1 |- ( ( F e. ( J Cn K ) /\ ( J ↾t A ) e. Comp ) -> ( K ↾t ( F " A ) ) e. Comp )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884  (class class class)co 6650   ↾_t crest 16081   Topctop 20698   Cn ccn 21028   Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190

This theorem is referenced by:  kgencn3  21361  txkgen  21455  xkoco1cn  21460  xkococnlem  21462  cmphaushmeo  21603  cnheiborlem  22753