Theorem cnntr 14545

Description: Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Assertion

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

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

Proof of Theorem cnntr

Step	Hyp	Ref	Expression
1		cnf2 14525	. . . 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 3615	. . . . . . 7 |- ( x e. ~P Y -> x C_ Y )
4	3	adantl 277	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ Y )
5		toponuni 14335	. . . . . . 7 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 489	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> Y = U. K )
7	4, 6	sseqtrd 3222	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ U. K )
8		eqid 2196	. . . . . . 7 |- U. K = U. K
9	8	cnntri 14544	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ x C_ U. K ) -> ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) )
10	9	expcom 116	. . . . 5 |- ( x C_ U. K -> ( F e. ( J Cn K ) -> ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
11	7, 10	syl 14	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> ( F e. ( J Cn K ) -> ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
12	11	ralrimdva 2577	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
13	2, 12	jcad 307	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> ( F : X --> Y /\ A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) ) )
14		toponss 14346	. . . . . . . . . 10 |- ( ( K e. (TopOn ` Y ) /\ x e. K ) -> x C_ Y )
15		velpw 3613	. . . . . . . . . 10 |- ( x e. ~P Y <-> x C_ Y )
16	14, 15	sylibr 134	. . . . . . . . 9 |- ( ( K e. (TopOn ` Y ) /\ x e. K ) -> x e. ~P Y )
17	16	ex 115	. . . . . . . 8 |- ( K e. (TopOn ` Y ) -> ( x e. K -> x e. ~P Y ) )
18	17	ad2antlr 489	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( x e. K -> x e. ~P Y ) )
19	18	imim1d 75	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( ( x e. ~P Y -> ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) -> ( x e. K -> ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) ) )
20		topontop 14334	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
21	20	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> J e. Top )
22		cnvimass 5033	. . . . . . . . . . 11 |- ( `' F " x ) C_ dom F
23		fdm 5416	. . . . . . . . . . . . 13 |- ( F : X --> Y -> dom F = X )
24	23	ad2antlr 489	. . . . . . . . . . . 12 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> dom F = X )
25		toponuni 14335	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) -> X = U. J )
26	25	ad3antrrr 492	. . . . . . . . . . . 12 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> X = U. J )
27	24, 26	eqtrd 2229	. . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> dom F = U. J )
28	22, 27	sseqtrid 3234	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( `' F " x ) C_ U. J )
29		eqid 2196	. . . . . . . . . . 11 |- U. J = U. J
30	29	ntrss2 14441	. . . . . . . . . 10 |- ( ( J e. Top /\ ( `' F " x ) C_ U. J ) -> ( ( int ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) )
31	21, 28, 30	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( int ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) )
32		eqss 3199	. . . . . . . . . 10 |- ( ( ( int ` J ) ` ( `' F " x ) ) = ( `' F " x ) <-> ( ( ( int ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) /\ ( `' F " x ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
33	32	baib 920	. . . . . . . . 9 |- ( ( ( int ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) -> ( ( ( int ` J ) ` ( `' F " x ) ) = ( `' F " x ) <-> ( `' F " x ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
34	31, 33	syl 14	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( ( int ` J ) ` ( `' F " x ) ) = ( `' F " x ) <-> ( `' F " x ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
35	29	isopn3 14445	. . . . . . . . 9 |- ( ( J e. Top /\ ( `' F " x ) C_ U. J ) -> ( ( `' F " x ) e. J <-> ( ( int ` J ) ` ( `' F " x ) ) = ( `' F " x ) ) )
36	21, 28, 35	syl2anc 411	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( `' F " x ) e. J <-> ( ( int ` J ) ` ( `' F " x ) ) = ( `' F " x ) ) )
37		topontop 14334	. . . . . . . . . . . 12 |- ( K e. (TopOn ` Y ) -> K e. Top )
38	37	ad3antlr 493	. . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> K e. Top )
39		isopn3i 14455	. . . . . . . . . . 11 |- ( ( K e. Top /\ x e. K ) -> ( ( int ` K ) ` x ) = x )
40	38, 39	sylancom 420	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( int ` K ) ` x ) = x )
41	40	imaeq2d 5010	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( `' F " ( ( int ` K ) ` x ) ) = ( `' F " x ) )
42	41	sseq1d 3213	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) <-> ( `' F " x ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
43	34, 36, 42	3bitr4rd 221	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) <-> ( `' F " x ) e. J ) )
44	43	pm5.74da 443	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( ( x e. K -> ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) <-> ( x e. K -> ( `' F " x ) e. J ) ) )
45	19, 44	sylibd 149	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( ( x e. ~P Y -> ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) -> ( x e. K -> ( `' F " x ) e. J ) ) )
46	45	ralimdv2 2567	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) -> ( A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) -> A. x e. K ( `' F " x ) e. J ) )
47	46	imdistanda 448	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( ( F : X --> Y /\ A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) -> ( F : X --> Y /\ A. x e. K ( `' F " x ) e. J ) ) )
48		iscn 14517	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. K ( `' F " x ) e. J ) ) )
49	47, 48	sylibrd 169	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( ( F : X --> Y /\ A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) -> F e. ( J Cn K ) ) )
50	13, 49	impbid 129	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   ~Pcpw 3606   U.cuni 3840   `'ccnv 4663   dom cdm 4664   "cima 4667   -->wf 5255   ` cfv 5259  (class class class)co 5925   Topctop 14317  TopOnctopon 14330   intcnt 14413   Cn ccn 14505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-top 14318  df-topon 14331  df-ntr 14416  df-cn 14508

This theorem is referenced by: (None)