Theorem cnpdis 15107

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 537	. . . . . . . 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 1065	. . . . . . . . 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 3718	. . . . . . . . 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 533	. . . . . . . . . 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 538	. . . . . . . . . . 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 5508	. . . . . . . . . . 11 |- ( f : X --> Y -> f Fn X )
8		elpreima 5797	. . . . . . . . . . 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 953	. . . . . . . . 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 3839	. . . . . . . 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 2296	. . . . . . . . . 10 |- ( y = { A } -> ( A e. y <-> A e. { A } ) )
13		sseq1 3261	. . . . . . . . . 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 2921	. . . . . . . 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 1272	. . . . . . 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 2615	. . . . 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 1028	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> K e. (TopOn ` Y ) )
22		toponmax 14890	. . . . 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 1027	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> J e. (TopOn ` X ) )
25		toponmax 14890	. . . . 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 6896	. . 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 15068	. . . 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 2230	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 1005   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   C_ wss 3211   {csn 3689   `'ccnv 4748   "cima 4752   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   ^m cmap 6882  TopOnctopon 14875   CnP ccnp 15051

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-top 14863  df-topon 14876  df-cnp 15054

This theorem is referenced by: (None)