Theorem cnclima 14946

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 14927	. . . . 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 5485	. . . . . 6 |- ( F : U. J --> U. K -> Fun F )
6		funcnvcnv 5389	. . . . . 6 |- ( Fun F -> Fun `' `' F )
7		imadif 5410	. . . . . 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 5776	. . . . . 6 |- ( F : U. J --> U. K -> ( `' F " U. K ) = U. J )
10	9	difeq1d 3324	. . . . 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 14830	. . . 4 |- ( A e. ( Clsd ` K ) -> ( U. K \ A ) e. K )
14		cnima 14943	. . . 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 14924	. . . 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 5099	. . . 4 |- ( `' F " A ) C_ dom F
20	19, 4	fssdm 5497	. . 3 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " A ) C_ U. J )
21	1	iscld2 14827	. . 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 1397   e. wcel 2202   \ cdif 3197   C_ wss 3200   U.cuni 3893   `'ccnv 4724   "cima 4728   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6017   Topctop 14720   Clsdccld 14815   Cn ccn 14908

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cld 14818  df-cn 14911

This theorem is referenced by:  hmeocld  15035