Theorem cndis 13035

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 4974	. . . . . . . 8 |- ( `' f " x ) C_ dom f
2		fdm 5353	. . . . . . . . 9 |- ( f : A --> X -> dom f = A )
3	2	adantl 275	. . . . . . . 8 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> dom f = A )
4	1, 3	sseqtrid 3197	. . . . . . 7 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( `' f " x ) C_ A )
5		elpw2g 4142	. . . . . . . 8 |- ( A e. V -> ( ( `' f " x ) e. ~P A <-> ( `' f " x ) C_ A ) )
6	5	ad2antrr 485	. . . . . . 7 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( ( `' f " x ) e. ~P A <-> ( `' f " x ) C_ A ) )
7	4, 6	mpbird 166	. . . . . 6 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> ( `' f " x ) e. ~P A )
8	7	ralrimivw 2544	. . . . 5 |- ( ( ( A e. V /\ J e. (TopOn ` X ) ) /\ f : A --> X ) -> A. x e. J ( `' f " x ) e. ~P A )
9	8	ex 114	. . . 4 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f : A --> X -> A. x e. J ( `' f " x ) e. ~P A ) )
10	9	pm4.71d 391	. . 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 12817	. . . 4 |- ( J e. (TopOn ` X ) -> X e. J )
12		id 19	. . . 4 |- ( A e. V -> A e. V )
13		elmapg 6639	. . . 4 |- ( ( X e. J /\ A e. V ) -> ( f e. ( X ^m A ) <-> f : A --> X ) )
14	11, 12, 13	syl2anr 288	. . 3 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f e. ( X ^m A ) <-> f : A --> X ) )
15		distopon 12881	. . . 4 |- ( A e. V -> ~P A e. (TopOn ` A ) )
16		iscn 12991	. . . 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 281	. . 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 220	. 2 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( f e. ( ~P A Cn J ) <-> f e. ( X ^m A ) ) )
19	18	eqrdv 2168	1 |- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( ~P A Cn J ) = ( X ^m A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   ~Pcpw 3566   `'ccnv 4610   dom cdm 4611   "cima 4614   -->wf 5194   ` cfv 5198  (class class class)co 5853   ^m cmap 6626  TopOnctopon 12802   Cn ccn 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-top 12790  df-topon 12803  df-cn 12982

This theorem is referenced by: (None)