Theorem cncls2 21077

Description: Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
cncls2	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. ~P Y ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) ) )

Distinct variable groups:   x, F   x, J   x, K   x, X   x, Y

Proof of Theorem cncls2

Step	Hyp	Ref	Expression
1		cnf2 21053	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
2	1	3expia 1267	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> F : X --> Y ) )
3		elpwi 4168	. . . . . . 7 |- ( x e. ~P Y -> x C_ Y )
4	3	adantl 482	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ Y )
5		toponuni 20719	. . . . . . 7 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 763	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> Y = U. K )
7	4, 6	sseqtrd 3641	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ U. K )
8		eqid 2622	. . . . . . 7 |- U. K = U. K
9	8	cncls2i 21074	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ x C_ U. K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) )
10	9	expcom 451	. . . . 5 |- ( x C_ U. K -> ( F e. ( J Cn K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) )
11	7, 10	syl 17	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> ( F e. ( J Cn K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) )
12	11	ralrimdva 2969	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> A. x e. ~P Y ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) )
13	2, 12	jcad 555	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> ( F : X --> Y /\ A. x e. ~P Y ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) ) )
14	8	cldss2 20834	. . . . . . . . 9 |- ( Clsd ` K ) C_ ~P U. K
15	5	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> Y = U. K )
16	15	pweqd 4163	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ~P Y = ~P U. K )
17	14, 16	syl5sseqr 3654	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( Clsd ` K ) C_ ~P Y )
18	17	sseld 3602	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( x e. ( Clsd ` K ) -> x e. ~P Y ) )
19	18	imim1d 82	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( ( x e. ~P Y -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) -> ( x e. ( Clsd ` K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) ) )
20		cldcls 20846	. . . . . . . . . . . 12 |- ( x e. ( Clsd ` K ) -> ( ( cls ` K ) ` x ) = x )
21	20	ad2antll 765	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( ( cls ` K ) ` x ) = x )
22	21	imaeq2d 5466	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( `' F " ( ( cls ` K ) ` x ) ) = ( `' F " x ) )
23	22	sseq2d 3633	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) <-> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) ) )
24		topontop 20718	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
25	24	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> J e. Top )
26		cnvimass 5485	. . . . . . . . . . 11 |- ( `' F " x ) C_ dom F
27		fdm 6051	. . . . . . . . . . . . 13 |- ( F : X --> Y -> dom F = X )
28	27	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = X )
29		toponuni 20719	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) -> X = U. J )
30	29	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> X = U. J )
31	28, 30	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = U. J )
32	26, 31	syl5sseq 3653	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( `' F " x ) C_ U. J )
33		eqid 2622	. . . . . . . . . . 11 |- U. J = U. J
34	33	iscld4 20869	. . . . . . . . . 10 |- ( ( J e. Top /\ ( `' F " x ) C_ U. J ) -> ( ( `' F " x ) e. ( Clsd ` J ) <-> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) ) )
35	25, 32, 34	syl2anc 693	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( ( `' F " x ) e. ( Clsd ` J ) <-> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) ) )
36	23, 35	bitr4d 271	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) <-> ( `' F " x ) e. ( Clsd ` J ) ) )
37	36	expr 643	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( x e. ( Clsd ` K ) -> ( ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) <-> ( `' F " x ) e. ( Clsd ` J ) ) ) )
38	37	pm5.74d 262	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( ( x e. ( Clsd ` K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) <-> ( x e. ( Clsd ` K ) -> ( `' F " x ) e. ( Clsd ` J ) ) ) )
39	19, 38	sylibd 229	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( ( x e. ~P Y -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) -> ( x e. ( Clsd ` K ) -> ( `' F " x ) e. ( Clsd ` J ) ) ) )
40	39	ralimdv2 2961	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( A. x e. ~P Y ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) -> A. x e. ( Clsd ` K ) ( `' F " x ) e. ( Clsd ` J ) ) )
41	40	imdistanda 729	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( ( F : X --> Y /\ A. x e. ~P Y ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) -> ( F : X --> Y /\ A. x e. ( Clsd ` K ) ( `' F " x ) e. ( Clsd ` J ) ) ) )
42		iscncl 21073	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. ( Clsd ` K ) ( `' F " x ) e. ( Clsd ` J ) ) ) )
43	41, 42	sylibrd 249	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( ( F : X --> Y /\ A. x e. ~P Y ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) -> F e. ( J Cn K ) ) )
44	13, 43	impbid 202	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. ~P Y ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   `'ccnv 5113   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   Clsdccld 20820   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