Theorem cnclima 15034

Description: A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnclima	|- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " A ) e. ( Clsd ` J ) )

Proof of Theorem cnclima

Step	Hyp	Ref	Expression
1		eqid 2231	. . . . . 6 |- U. J = U. J
2		eqid 2231	. . . . . 6 |- U. K = U. K
3	1, 2	cnf 15015	. . . . 5 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
4	3	adantr 276	. . . 4 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> F : U. J --> U. K )
5		ffun 5492	. . . . . 6 |- ( F : U. J --> U. K -> Fun F )
6		funcnvcnv 5396	. . . . . 6 |- ( Fun F -> Fun `' `' F )
7		imadif 5417	. . . . . 6 |- ( Fun `' `' F -> ( `' F " ( U. K \ A ) ) = ( ( `' F " U. K ) \ ( `' F " A ) ) )
8	5, 6, 7	3syl 17	. . . . 5 |- ( F : U. J --> U. K -> ( `' F " ( U. K \ A ) ) = ( ( `' F " U. K ) \ ( `' F " A ) ) )
9		fimacnv 5784	. . . . . 6 |- ( F : U. J --> U. K -> ( `' F " U. K ) = U. J )
10	9	difeq1d 3326	. . . . 5 |- ( F : U. J --> U. K -> ( ( `' F " U. K ) \ ( `' F " A ) ) = ( U. J \ ( `' F " A ) ) )
11	8, 10	eqtr2d 2265	. . . 4 |- ( F : U. J --> U. K -> ( U. J \ ( `' F " A ) ) = ( `' F " ( U. K \ A ) ) )
12	4, 11	syl 14	. . 3 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( U. J \ ( `' F " A ) ) = ( `' F " ( U. K \ A ) ) )
13	2	cldopn 14918	. . . 4 |- ( A e. ( Clsd ` K ) -> ( U. K \ A ) e. K )
14		cnima 15031	. . . 4 |- ( ( F e. ( J Cn K ) /\ ( U. K \ A ) e. K ) -> ( `' F " ( U. K \ A ) ) e. J )
15	13, 14	sylan2 286	. . 3 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " ( U. K \ A ) ) e. J )
16	12, 15	eqeltrd 2308	. 2 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( U. J \ ( `' F " A ) ) e. J )
17		cntop1 15012	. . . 4 |- ( F e. ( J Cn K ) -> J e. Top )
18	17	adantr 276	. . 3 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> J e. Top )
19		cnvimass 5106	. . . 4 |- ( `' F " A ) C_ dom F
20	19, 4	fssdm 5504	. . 3 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " A ) C_ U. J )
21	1	iscld2 14915	. . 3 |- ( ( J e. Top /\ ( `' F " A ) C_ U. J ) -> ( ( `' F " A ) e. ( Clsd ` J ) <-> ( U. J \ ( `' F " A ) ) e. J ) )
22	18, 20, 21	syl2anc 411	. 2 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( ( `' F " A ) e. ( Clsd ` J ) <-> ( U. J \ ( `' F " A ) ) e. J ) )
23	16, 22	mpbird 167	1 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " A ) e. ( Clsd ` J ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   \ cdif 3198   C_ wss 3201   U.cuni 3898   `'ccnv 4730   "cima 4734   Fun wfun 5327   -->wf 5329   ` cfv 5333  (class class class)co 6028   Topctop 14808   Clsdccld 14903   Cn ccn 14996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-top 14809  df-topon 14822  df-cld 14906  df-cn 14999

This theorem is referenced by:  hmeocld  15123