Theorem cncls2 20276

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 20252	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
2	1	3expia 1207	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> F : X --> Y ) )
3		elpwi 3988	. . . . . . 7 |- ( x e. ~P Y -> x C_ Y )
4	3	adantl 467	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ Y )
5		toponuni 19929	. . . . . . 7 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 731	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> Y = U. K )
7	4, 6	sseqtrd 3500	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ U. K )
8		eqid 2422	. . . . . . 7 |- U. K = U. K
9	8	cncls2i 20273	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ x C_ U. K ) -> ( ( cls ` J ) ` ( `' F " x ) ) C_ ( `' F " ( ( cls ` K ) ` x ) ) )
10	9	expcom 436	. . . . 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 2843	. . 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 535	. 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 20032	. . . . . . . . 9 |- ( Clsd ` K ) C_ ~P U. K
15	5	ad2antlr 731	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> Y = U. K )
16	15	pweqd 3984	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ~P Y = ~P U. K )
17	14, 16	syl5sseqr 3513	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( Clsd ` K ) C_ ~P Y )
18	17	sseld 3463	. . . . . . 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 20044	. . . . . . . . . . . 12 |- ( x e. ( Clsd ` K ) -> ( ( cls ` K ) ` x ) = x )
21	20	ad2antll 733	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( ( cls ` K ) ` x ) = x )
22	21	imaeq2d 5184	. . . . . . . . . 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 3492	. . . . . . . . 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 19928	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
25	24	ad2antrr 730	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> J e. Top )
26		cnvimass 5204	. . . . . . . . . . 11 |- ( `' F " x ) C_ dom F
27		fdm 5747	. . . . . . . . . . . . 13 |- ( F : X --> Y -> dom F = X )
28	27	ad2antrl 732	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = X )
29		toponuni 19929	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) -> X = U. J )
30	29	ad2antrr 730	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> X = U. J )
31	28, 30	eqtrd 2463	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> dom F = U. J )
32	26, 31	syl5sseq 3512	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( F : X --> Y /\ x e. ( Clsd ` K ) ) ) -> ( `' F " x ) C_ U. J )
33		eqid 2422	. . . . . . . . . . 11 |- U. J = U. J
34	33	iscld4 20068	. . . . . . . . . 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 665	. . . . . . . . 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 259	. . . . . . . 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 618	. . . . . . 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 250	. . . . . 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 217	. . . . 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 2832	. . . 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 697	. . 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 20272	. . 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 237	. 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 193	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 187   /\ wa 370   = wceq 1437   e. wcel 1868   A.wral 2775   C_ wss 3436   ~Pcpw 3979   U.cuni 4216   `'ccnv 4849   dom cdm 4850   "cima 4853   -->wf 5594   ` cfv 5598  (class class class)co 6302   Topctop 19904  TopOnctopon 19905   Clsdccld 20018   clsccl 20020   Cn ccn 20227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-map 7479  df-top 19908  df-topon 19910  df-cld 20021  df-cls 20023  df-cn 20230

This theorem is referenced by:  cncls  20277