Theorem cnpdis 13409

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 1038	. . . . . . . . 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 3620	. . . . . . . . 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 5361	. . . . . . . . . . 11 |- ( f : X --> Y -> f Fn X )
8		elpreima 5631	. . . . . . . . . . 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 944	. . . . . . . . 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 3736	. . . . . . . 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 2241	. . . . . . . . . 10 |- ( y = { A } -> ( A e. y <-> A e. { A } ) )
13		sseq1 3178	. . . . . . . . . 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 2841	. . . . . . . 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 1236	. . . . . . 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 2550	. . . . 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 1001	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> K e. (TopOn ` Y ) )
22		toponmax 13190	. . . . 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 1000	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> J e. (TopOn ` X ) )
25		toponmax 13190	. . . . 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 6656	. . 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 13370	. . . 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 2175	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 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   C_ wss 3129   {csn 3591   `'ccnv 4622   "cima 4626   Fn wfn 5207   -->wf 5208   ` cfv 5212  (class class class)co 5869   ^m cmap 6642  TopOnctopon 13175   CnP ccnp 13353

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-top 13163  df-topon 13176  df-cnp 13356

This theorem is referenced by: (None)