Theorem cncls2 20289

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 20265	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
2	1	3expia 1210	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> F : X --> Y ) )
3		elpwi 3960	. . . . . . 7 |- ( x e. ~P Y -> x C_ Y )
4	3	adantl 468	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ Y )
5		toponuni 19942	. . . . . . 7 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 733	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> Y = U. K )
7	4, 6	sseqtrd 3468	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ U. K )
8		eqid 2451	. . . . . . 7 |- U. K = U. K
9	8	cncls2i 20286	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ x C_ U. K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) )
10	9	expcom 437	. . . . 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 2806	. . 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 536	. 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 20045	. . . . . . . . 9 |- ( Clsd ` K ) C_ ~P U. K
15	5	ad2antlr 733	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> Y = U. K )
16	15	pweqd 3956	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ~P Y = ~P U. K )
17	14, 16	syl5sseqr 3481	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( Clsd ` K ) C_ ~P Y )
18	17	sseld 3431	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( x e. ( Clsd ` K ) -> x e. ~P Y ) )
19	18	imim1d 78	. . . . . 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 20057	. . . . . . . . . . . 12 |- ( x e. ( Clsd ` K ) -> ( ( cls ` K ) ` x ) = x )
21	20	ad2antll 735	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( ( cls ` K ) ` x ) = x )
22	21	imaeq2d 5168	. . . . . . . . . 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 3460	. . . . . . . . 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 19941	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
25	24	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> J e. Top )
26		cnvimass 5188	. . . . . . . . . . 11 |- ( `' F " x ) C_ dom F
27		fdm 5733	. . . . . . . . . . . . 13 |- ( F : X --> Y -> dom F = X )
28	27	ad2antrl 734	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = X )
29		toponuni 19942	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) -> X = U. J )
30	29	ad2antrr 732	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> X = U. J )
31	28, 30	eqtrd 2485	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = U. J )
32	26, 31	syl5sseq 3480	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( `' F " x ) C_ U. J )
33		eqid 2451	. . . . . . . . . . 11 |- U. J = U. J
34	33	iscld4 20081	. . . . . . . . . 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 667	. . . . . . . . 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 260	. . . . . . . 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 620	. . . . . . 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 251	. . . . . 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 218	. . . . 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 2795	. . . 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 699	. . 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 20285	. . 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 238	. 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 194	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 188   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   C_ wss 3404   ~Pcpw 3951   U.cuni 4198   `'ccnv 4833   dom cdm 4834   "cima 4837   -->wf 5578   ` cfv 5582  (class class class)co 6290   Topctop 19917  TopOnctopon 19918   Clsdccld 20031   clsccl 20033   Cn ccn 20240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-map 7474  df-top 19921  df-topon 19923  df-cld 20034  df-cls 20036  df-cn 20243

This theorem is referenced by:  cncls  20290