Theorem cncls2 20366

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 20342	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
2	1	3expia 1233	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> F : X --> Y ) )
3		elpwi 3951	. . . . . . 7 |- ( x e. ~P Y -> x C_ Y )
4	3	adantl 473	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ Y )
5		toponuni 20019	. . . . . . 7 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 741	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> Y = U. K )
7	4, 6	sseqtrd 3454	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ U. K )
8		eqid 2471	. . . . . . 7 |- U. K = U. K
9	8	cncls2i 20363	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ x C_ U. K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) )
10	9	expcom 442	. . . . 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 2812	. . 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 542	. 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 20122	. . . . . . . . 9 |- ( Clsd ` K ) C_ ~P U. K
15	5	ad2antlr 741	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> Y = U. K )
16	15	pweqd 3947	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ~P Y = ~P U. K )
17	14, 16	syl5sseqr 3467	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( Clsd ` K ) C_ ~P Y )
18	17	sseld 3417	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( x e. ( Clsd ` K ) -> x e. ~P Y ) )
19	18	imim1d 77	. . . . . 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 20134	. . . . . . . . . . . 12 |- ( x e. ( Clsd ` K ) -> ( ( cls ` K ) ` x ) = x )
21	20	ad2antll 743	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( ( cls ` K ) ` x ) = x )
22	21	imaeq2d 5174	. . . . . . . . . 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 3446	. . . . . . . . 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 20018	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
25	24	ad2antrr 740	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> J e. Top )
26		cnvimass 5194	. . . . . . . . . . 11 |- ( `' F " x ) C_ dom F
27		fdm 5745	. . . . . . . . . . . . 13 |- ( F : X --> Y -> dom F = X )
28	27	ad2antrl 742	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = X )
29		toponuni 20019	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) -> X = U. J )
30	29	ad2antrr 740	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> X = U. J )
31	28, 30	eqtrd 2505	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = U. J )
32	26, 31	syl5sseq 3466	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( `' F " x ) C_ U. J )
33		eqid 2471	. . . . . . . . . . 11 |- U. J = U. J
34	33	iscld4 20158	. . . . . . . . . 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 673	. . . . . . . . 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 264	. . . . . . . 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 626	. . . . . . 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 255	. . . . . 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 222	. . . . 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 2804	. . . 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 707	. . 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 20362	. . 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 242	. 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 195	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   `'ccnv 4838   dom cdm 4839   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308   Topctop 19994  TopOnctopon 19995   Clsdccld 20108   clsccl 20110   Cn ccn 20317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-top 19998  df-topon 20000  df-cld 20111  df-cls 20113  df-cn 20320

This theorem is referenced by:  cncls  20367