Theorem cndis 15052

Description: Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cndis	|- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( ~P A Cn J ) = ( X ^m A ) )

Proof of Theorem cndis

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5106	. . . . . . . 8 |- ( `' f " x ) C_ dom f
2		fdm 5495	. . . . . . . . 9 |- ( f : A --> X -> dom f = A )
3	2	adantl 277	. . . . . . . 8 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> dom f = A )
4	1, 3	sseqtrid 3278	. . . . . . 7 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( `' f " x ) C_ A )
5		elpw2g 4251	. . . . . . . 8 |- ( A e. V -> ( ( `' f " x ) e. ~P A <-> ( `' f " x ) C_ A ) )
6	5	ad2antrr 488	. . . . . . 7 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( ( `' f " x ) e. ~P A <-> ( `' f " x ) C_ A ) )
7	4, 6	mpbird 167	. . . . . 6 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( `' f " x ) e. ~P A )
8	7	ralrimivw 2607	. . . . 5 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> A. x e. J ( `' f " x ) e. ~P A )
9	8	ex 115	. . . 4 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f : A --> X -> A. x e. J ( `' f " x ) e. ~P A ) )
10	9	pm4.71d 393	. . 3 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f : A --> X <-> ( f : A --> X /\ A. x e. J ( `' f " x ) e. ~P A ) ) )
11		toponmax 14836	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
12		id 19	. . . 4 |- ( A e. V -> A e. V )
13		elmapg 6873	. . . 4 |- ( ( X e. J /\ A e. V ) -> ( f e. ( X ^m A ) <-> f : A --> X ) )
14	11, 12, 13	syl2anr 290	. . 3 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f e. ( X ^m A ) <-> f : A --> X ) )
15		distopon 14898	. . . 4 |- ( A e. V -> ~P A e. (TopOn ` A ) )
16		iscn 15008	. . . 4 |- ( ( ~P A e. (TopOn ` A ) /\ J e. (TopOn ` X ) ) -> ( f e. ( ~P A Cn J ) <-> ( f : A --> X /\ A. x e. J ( `' f " x ) e. ~P A ) ) )
17	15, 16	sylan 283	. . 3 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f e. ( ~P A Cn J ) <-> ( f : A --> X /\ A. x e. J ( `' f " x ) e. ~P A ) ) )
18	10, 14, 17	3bitr4rd 221	. 2 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f e. ( ~P A Cn J ) <-> f e. ( X ^m A ) ) )
19	18	eqrdv 2229	1 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( ~P A Cn J ) = ( X ^m A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   ~Pcpw 3656   `'ccnv 4730   dom cdm 4731   "cima 4734   -->wf 5329   ` cfv 5333  (class class class)co 6028   ^m cmap 6860  TopOnctopon 14821   Cn ccn 14996

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-top 14809  df-topon 14822  df-cn 14999

This theorem is referenced by: (None)