Theorem cndis 15106

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 5125	. . . . . . . 8 |- ( `' f " x ) C_ dom f
2		fdm 5514	. . . . . . . . 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 3288	. . . . . . 7 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( `' f " x ) C_ A )
5		elpw2g 4268	. . . . . . . 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 2616	. . . . 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 14890	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
12		id 19	. . . 4 |- ( A e. V -> A e. V )
13		elmapg 6895	. . . 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 14952	. . . 4 |- ( A e. V -> ~P A e. (TopOn ` A ) )
16		iscn 15062	. . . 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 2230	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 2203   A.wral 2520   C_ wss 3211   ~Pcpw 3669   `'ccnv 4748   dom cdm 4749   "cima 4752   -->wf 5348   ` cfv 5352  (class class class)co 6050   ^m cmap 6882  TopOnctopon 14875   Cn ccn 15050

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-top 14863  df-topon 14876  df-cn 15053

This theorem is referenced by: (None)