Theorem cnntr 15326

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 15306	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
2	1	3expia 1236	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) -> F : X --> Y ) )
3		elpwi 3698	. . . . . . 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 15116	. . . . . . 7 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 493	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> Y = U. K )
7	4, 6	sseqtrd 3286	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ x e. ~P Y ) -> x C_ U. K )
8		eqid 2238	. . . . . . 7 |- U. K = U. K
9	8	cnntri 15325	. . . . . 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 2630	. . 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 15127	. . . . . . . . . 10 |- ( ( K e. (TopOn ` Y ) /\ x e. K ) -> x C_ Y )
15		velpw 3695	. . . . . . . . . 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 493	. . . . . . 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 15115	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
21	20	ad3antrrr 496	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> J e. Top )
22		cnvimass 5150	. . . . . . . . . . 11 |- ( `' F " x ) C_ dom F
23		fdm 5539	. . . . . . . . . . . . 13 |- ( F : X --> Y -> dom F = X )
24	23	ad2antlr 493	. . . . . . . . . . . 12 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> dom F = X )
25		toponuni 15116	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) -> X = U. J )
26	25	ad3antrrr 496	. . . . . . . . . . . 12 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> X = U. J )
27	24, 26	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> dom F = U. J )
28	22, 27	sseqtrid 3298	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( `' F " x ) C_ U. J )
29		eqid 2238	. . . . . . . . . . 11 |- U. J = U. J
30	29	ntrss2 15222	. . . . . . . . . 10 |- ( ( J e. Top /\ ( `' F " x ) C_ U. J ) -> ( ( int ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) )
31	21, 28, 30	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( int ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) )
32		eqss 3263	. . . . . . . . . 10 |- ( ( ( int ` J ) ` ( `' F " x ) ) = ( `' F " x ) <-> ( ( ( int ` J ) ` ( `' F " x ) ) C_ ( `' F " x ) /\ ( `' F " x ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) )
33	32	baib 931	. . . . . . . . 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 15226	. . . . . . . . 9 |- ( ( J e. Top /\ ( `' F " x ) C_ U. J ) -> ( ( `' F " x ) e. J <-> ( ( int ` J ) ` ( `' F " x ) ) = ( `' F " x ) ) )
36	21, 28, 35	syl2anc 415	. . . . . . . 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 15115	. . . . . . . . . . . 12 |- ( K e. (TopOn ` Y ) -> K e. Top )
38	37	ad3antlr 497	. . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> K e. Top )
39		isopn3i 15236	. . . . . . . . . . 11 |- ( ( K e. Top /\ x e. K ) -> ( ( int ` K ) ` x ) = x )
40	38, 39	sylancom 424	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( ( int ` K ) ` x ) = x )
41	40	imaeq2d 5126	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F : X --> Y ) /\ x e. K ) -> ( `' F " ( ( int ` K ) ` x ) ) = ( `' F " x ) )
42	41	sseq1d 3277	. . . . . . . 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 447	. . . . . 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 2620	. . . 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 452	. . 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 15298	. . 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 ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ~Pcpw 3688   U.cuni 3935   `'ccnv 4773   dom cdm 4774   "cima 4777   -->wf 5373   ` cfv 5377  (class class class)co 6085   Topctop 15098  TopOnctopon 15111   intcnt 15194   Cn ccn 15286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-top 15099  df-topon 15112  df-ntr 15197  df-cn 15289

This theorem is used by: (None)