ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnsscnp Unicode version

Theorem cnsscnp 14126
Description: The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnsscnp.1  |-  X  = 
U. J
Assertion
Ref Expression
cnsscnp  |-  ( P  e.  X  ->  ( J  Cn  K )  C_  ( ( J  CnP  K ) `  P ) )

Proof of Theorem cnsscnp
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cnsscnp.1 . . . 4  |-  X  = 
U. J
21cncnpi 14125 . . 3  |-  ( ( f  e.  ( J  Cn  K )  /\  P  e.  X )  ->  f  e.  ( ( J  CnP  K ) `
 P ) )
32expcom 116 . 2  |-  ( P  e.  X  ->  (
f  e.  ( J  Cn  K )  -> 
f  e.  ( ( J  CnP  K ) `
 P ) ) )
43ssrdv 3176 1  |-  ( P  e.  X  ->  ( J  Cn  K )  C_  ( ( J  CnP  K ) `  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160    C_ wss 3144   U.cuni 3824   ` cfv 5231  (class class class)co 5891    Cn ccn 14082    CnP ccnp 14083
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-map 6668  df-top 13895  df-topon 13908  df-cn 14085  df-cnp 14086
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator