Theorem cnclima 14810

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 2207	. . . . . 6 |- U. J = U. J
2		eqid 2207	. . . . . 6 |- U. K = U. K
3	1, 2	cnf 14791	. . . . 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 5448	. . . . . 6 |- ( F : U. J --> U. K -> Fun F )
6		funcnvcnv 5352	. . . . . 6 |- ( Fun F -> Fun `' `' F )
7		imadif 5373	. . . . . 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 5732	. . . . . 6 |- ( F : U. J --> U. K -> ( `' F " U. K ) = U. J )
10	9	difeq1d 3298	. . . . 5 |- ( F : U. J --> U. K -> ( ( `' F " U. K ) \ ( `' F " A ) ) = ( U. J \ ( `' F " A ) ) )
11	8, 10	eqtr2d 2241	. . . 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 14694	. . . 4 |- ( A e. ( Clsd ` K ) -> ( U. K \ A ) e. K )
14		cnima 14807	. . . 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 2284	. 2 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( U. J \ ( `' F " A ) ) e. J )
17		cntop1 14788	. . . 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 5064	. . . 4 |- ( `' F " A ) C_ dom F
20	19, 4	fssdm 5460	. . 3 |- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " A ) C_ U. J )
21	1	iscld2 14691	. . 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 1373   e. wcel 2178   \ cdif 3171   C_ wss 3174   U.cuni 3864   `'ccnv 4692   "cima 4696   Fun wfun 5284   -->wf 5286   ` cfv 5290  (class class class)co 5967   Topctop 14584   Clsdccld 14679   Cn ccn 14772

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-top 14585  df-topon 14598  df-cld 14682  df-cn 14775

This theorem is referenced by:  hmeocld  14899