Theorem cnpdis 14916

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 1062	. . . . . . . . 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 3695	. . . . . . . . 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 5473	. . . . . . . . . . 11 |- ( f : X --> Y -> f Fn X )
8		elpreima 5754	. . . . . . . . . . 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 950	. . . . . . . . 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 3813	. . . . . . . 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 2293	. . . . . . . . . 10 |- ( y = { A } -> ( A e. y <-> A e. { A } ) )
13		sseq1 3247	. . . . . . . . . 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 2907	. . . . . . . 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 1269	. . . . . . 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 2603	. . . . 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 1025	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> K e. (TopOn ` Y ) )
22		toponmax 14699	. . . . 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 1024	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> J e. (TopOn ` X ) )
25		toponmax 14699	. . . . 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 6809	. . 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 14877	. . . 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 2227	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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   C_ wss 3197   {csn 3666   `'ccnv 4718   "cima 4722   Fn wfn 5313   -->wf 5314   ` cfv 5318  (class class class)co 6001   ^m cmap 6795  TopOnctopon 14684   CnP ccnp 14860

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-top 14672  df-topon 14685  df-cnp 14863

This theorem is referenced by: (None)