Theorem cnpdis 12450

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 525	. . . . . . . 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 1023	. . . . . . . . 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 3561	. . . . . . . . 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 522	. . . . . . . . . 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 526	. . . . . . . . . . 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 5280	. . . . . . . . . . 11 |- ( f : X --> Y -> f Fn X )
8		elpreima 5547	. . . . . . . . . . 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 929	. . . . . . . . 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 3673	. . . . . . . 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 2204	. . . . . . . . . 10 |- ( y = { A } -> ( A e. y <-> A e. { A } ) )
13		sseq1 3125	. . . . . . . . . 10 |- ( y = { A } -> ( y C_ ( `' f " x ) <-> { A } C_ ( `' f " x ) ) )
14	12, 13	anbi12d 465	. . . . . . . . 9 |- ( y = { A } -> ( ( A e. y /\ y C_ ( `' f " x ) ) <-> ( A e. { A } /\ { A } C_ ( `' f " x ) ) ) )
15	14	rspcev 2793	. . . . . . . 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 1215	. . . . . . 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 373	. . . . . 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 2508	. . . . 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 373	. . . 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 391	. . 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 986	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> K e. (TopOn ` Y ) )
22		toponmax 12231	. . . . 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 985	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> J e. (TopOn ` X ) )
25		toponmax 12231	. . . . 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 6564	. . 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 12411	. . . 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 274	. . 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 220	. 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 2138	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 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   C_ wss 3076   {csn 3532   `'ccnv 4546   "cima 4550   Fn wfn 5126   -->wf 5127   ` cfv 5131  (class class class)co 5782   ^m cmap 6550  TopOnctopon 12216   CnP ccnp 12394

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-top 12204  df-topon 12217  df-cnp 12397

This theorem is referenced by: (None)