Theorem cnclsi 21076

Description: Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
cnclsi.1	|- X = U. J

Assertion

Ref	Expression
cnclsi	|- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( F " ( ( cls ` J ) ` S ) ) C_ ( ( cls ` K ) ` ( F " S ) ) )

Proof of Theorem cnclsi

Step	Hyp	Ref	Expression
1		cntop1 21044	. . . . 5 |- ( F e. ( J Cn K ) -> J e. Top )
2	1	adantr 481	. . . 4 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> J e. Top )
3		cnvimass 5485	. . . . 5 |- ( `' F " ( F " S ) ) C_ dom F
4		cnclsi.1	. . . . . . . 8 |- X = U. J
5		eqid 2622	. . . . . . . 8 |- U. K = U. K
6	4, 5	cnf 21050	. . . . . . 7 |- ( F e. ( J Cn K ) -> F : X --> U. K )
7	6	adantr 481	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> F : X --> U. K )
8		fdm 6051	. . . . . 6 |- ( F : X --> U. K -> dom F = X )
9	7, 8	syl 17	. . . . 5 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> dom F = X )
10	3, 9	syl5sseq 3653	. . . 4 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( `' F " ( F " S ) ) C_ X )
11		simpr 477	. . . . . . 7 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> S C_ X )
12	11, 9	sseqtr4d 3642	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> S C_ dom F )
13		sseqin2 3817	. . . . . 6 |- ( S C_ dom F <-> ( dom F i^i S ) = S )
14	12, 13	sylib 208	. . . . 5 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( dom F i^i S ) = S )
15		dminss 5547	. . . . 5 |- ( dom F i^i S ) C_ ( `' F " ( F " S ) )
16	14, 15	syl6eqssr 3656	. . . 4 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> S C_ ( `' F " ( F " S ) ) )
17	4	clsss 20858	. . . 4 |- ( ( J e. Top /\ ( `' F " ( F " S ) ) C_ X /\ S C_ ( `' F " ( F " S ) ) ) -> ( ( cls ` J ) ` S ) C_ ( ( cls ` J ) ` ( `' F " ( F " S ) ) ) )
18	2, 10, 16, 17	syl3anc 1326	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ ( ( cls ` J ) ` ( `' F " ( F " S ) ) ) )
19		imassrn 5477	. . . . 5 |- ( F " S ) C_ ran F
20		frn 6053	. . . . . 6 |- ( F : X --> U. K -> ran F C_ U. K )
21	7, 20	syl 17	. . . . 5 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ran F C_ U. K )
22	19, 21	syl5ss 3614	. . . 4 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( F " S ) C_ U. K )
23	5	cncls2i 21074	. . . 4 |- ( ( F e. ( J Cn K ) /\ ( F " S ) C_ U. K ) -> ( ( cls ` J ) ` ( `' F " ( F " S ) ) ) C_ ( `' F " ( ( cls ` K ) ` ( F " S ) ) ) )
24	22, 23	syldan 487	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( ( cls ` J ) ` ( `' F " ( F " S ) ) ) C_ ( `' F " ( ( cls ` K ) ` ( F " S ) ) ) )
25	18, 24	sstrd 3613	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ ( `' F " ( ( cls ` K ) ` ( F " S ) ) ) )
26		ffun 6048	. . . 4 |- ( F : X --> U. K -> Fun F )
27	7, 26	syl 17	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> Fun F )
28	4	clsss3 20863	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
29	1, 28	sylan 488	. . . 4 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
30	29, 9	sseqtr4d 3642	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ dom F )
31		funimass3 6333	. . 3 |- ( ( Fun F /\ ( ( cls ` J ) ` S ) C_ dom F ) -> ( ( F " ( ( cls ` J ) ` S ) ) C_ ( ( cls ` K ) ` ( F " S ) ) <-> ( ( cls ` J ) ` S ) C_ ( `' F " ( ( cls ` K ) ` ( F " S ) ) ) ) )
32	27, 30, 31	syl2anc 693	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( ( F " ( ( cls ` J ) ` S ) ) C_ ( ( cls ` K ) ` ( F " S ) ) <-> ( ( cls ` J ) ` S ) C_ ( `' F " ( ( cls ` K ) ` ( F " S ) ) ) ) )
33	25, 32	mpbird 247	1 |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( F " ( ( cls ` J ) ` S ) ) C_ ( ( cls ` K ) ` ( F " S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   U.cuni 4436   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698   clsccl 20822   Cn ccn 21028

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-map 7859  df-top 20699  df-topon 20716  df-cld 20823  df-cls 20825  df-cn 21031

This theorem is referenced by:  cncls  21078  hmeocls  21571  clsnsg  21913