Theorem cnpdis 14410

Description: If A is an isolated point in X (or equivalently, the singleton { A } is open in X), then every function is continuous at A. (Contributed by Mario Carneiro, 9-Sep-2015.)

Assertion

Ref	Expression
cnpdis	|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( ( J CnP K ) ` A ) = ( Y ^m X ) )

Proof of Theorem cnpdis

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

Step	Hyp	Ref	Expression
1		simplrl 535	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> { A } e. J )
2		simpll3 1040	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> A e. X )
3		snidg 3647	. . . . . . . . 9 |- ( A e. X -> A e. { A } )
4	2, 3	syl 14	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> A e. { A } )
5		simprr 531	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> ( f ` A ) e. x )
6		simplrr 536	. . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> f : X --> Y )
7		ffn 5403	. . . . . . . . . . 11 |- ( f : X --> Y -> f Fn X )
8		elpreima 5677	. . . . . . . . . . 11 |- ( f Fn X -> ( A e. ( `' f " x ) <-> ( A e. X /\ ( f ` A ) e. x ) ) )
9	6, 7, 8	3syl 17	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> ( A e. ( `' f " x ) <-> ( A e. X /\ ( f ` A ) e. x ) ) )
10	2, 5, 9	mpbir2and 946	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> A e. ( `' f " x ) )
11	10	snssd 3763	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> { A } C_ ( `' f " x ) )
12		eleq2 2257	. . . . . . . . . 10 |- ( y = { A } -> ( A e. y <-> A e. { A } ) )
13		sseq1 3202	. . . . . . . . . 10 |- ( y = { A } -> ( y C_ ( `' f " x ) <-> { A } C_ ( `' f " x ) ) )
14	12, 13	anbi12d 473	. . . . . . . . 9 |- ( y = { A } -> ( ( A e. y /\ y C_ ( `' f " x ) ) <-> ( A e. { A } /\ { A } C_ ( `' f " x ) ) ) )
15	14	rspcev 2864	. . . . . . . 8 |- ( ( { A } e. J /\ ( A e. { A } /\ { A } C_ ( `' f " x ) ) ) -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) )
16	1, 4, 11, 15	syl12anc 1247	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ ( x e. K /\ ( f ` A ) e. x ) ) -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) )
17	16	expr 375	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) /\ x e. K ) -> ( ( f ` A ) e. x -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) ) )
18	17	ralrimiva 2567	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ ( { A } e. J /\ f : X --> Y ) ) -> A. x e. K ( ( f ` A ) e. x -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) ) )
19	18	expr 375	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( f : X --> Y -> A. x e. K ( ( f ` A ) e. x -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) ) ) )
20	19	pm4.71d 393	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( f : X --> Y <-> ( f : X --> Y /\ A. x e. K ( ( f ` A ) e. x -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) ) ) ) )
21		simpl2 1003	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> K e. (TopOn ` Y ) )
22		toponmax 14193	. . . . 5 |- ( K e. (TopOn ` Y ) -> Y e. K )
23	21, 22	syl 14	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> Y e. K )
24		simpl1 1002	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> J e. (TopOn ` X ) )
25		toponmax 14193	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
26	24, 25	syl 14	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> X e. J )
27	23, 26	elmapd 6716	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( f e. ( Y ^m X ) <-> f : X --> Y ) )
28		iscnp3 14371	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( f e. ( ( J CnP K ) ` A ) <-> ( f : X --> Y /\ A. x e. K ( ( f ` A ) e. x -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) ) ) ) )
29	28	adantr 276	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( f e. ( ( J CnP K ) ` A ) <-> ( f : X --> Y /\ A. x e. K ( ( f ` A ) e. x -> E. y e. J ( A e. y /\ y C_ ( `' f " x ) ) ) ) ) )
30	20, 27, 29	3bitr4rd 221	. 2 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( f e. ( ( J CnP K ) ` A ) <-> f e. ( Y ^m X ) ) )
31	30	eqrdv 2191	1 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( ( J CnP K ) ` A ) = ( Y ^m X ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   C_ wss 3153   {csn 3618   `'ccnv 4658   "cima 4662   Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   ^m cmap 6702  TopOnctopon 14178   CnP ccnp 14354

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-top 14166  df-topon 14179  df-cnp 14357

This theorem is referenced by: (None)