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Theorem cnsscnp 13956
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 13955 . . 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 3173 1  |-  ( P  e.  X  ->  ( J  Cn  K )  C_  ( ( J  CnP  K ) `  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158    C_ wss 3141   U.cuni 3821   ` cfv 5228  (class class class)co 5888    Cn ccn 13912    CnP ccnp 13913
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-map 6663  df-top 13725  df-topon 13738  df-cn 13915  df-cnp 13916
This theorem is referenced by: (None)
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