Theorem cndis 15342

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 5150	. . . . . . . 8 |- ( `' f " x ) C_ dom f
2		fdm 5539	. . . . . . . . 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 3298	. . . . . . 7 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( `' f " x ) C_ A )
5		elpw2g 4292	. . . . . . . 8 |- ( A e. V -> ( ( `' f " x ) e. ~P A <-> ( `' f " x ) C_ A ) )
6	5	ad2antrr 492	. . . . . . 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 2624	. . . . 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 397	. . 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 15126	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
12		id 19	. . . 4 |- ( A e. V -> A e. V )
13		elmapg 6935	. . . 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 15188	. . . 4 |- ( A e. V -> ~P A e. (TopOn ` A ) )
16		iscn 15298	. . . 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 2236	1 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( ~P A Cn J ) = ( X ^m A ) )

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   `'ccnv 4773   dom cdm 4774   "cima 4777   -->wf 5373   ` cfv 5377  (class class class)co 6085   ^m cmap 6922  TopOnctopon 15111   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-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-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-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-cn 15289

This theorem is used by: (None)