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Theorem cnsscnp 17345
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 17344 . . 3  |-  ( ( f  e.  ( J  Cn  K )  /\  P  e.  X )  ->  f  e.  ( ( J  CnP  K ) `
 P ) )
32expcom 426 . 2  |-  ( P  e.  X  ->  (
f  e.  ( J  Cn  K )  -> 
f  e.  ( ( J  CnP  K ) `
 P ) ) )
43ssrdv 3356 1  |-  ( P  e.  X  ->  ( J  Cn  K )  C_  ( ( J  CnP  K ) `  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    C_ wss 3322   U.cuni 4017   ` cfv 5456  (class class class)co 6083    Cn ccn 17290    CnP ccnp 17291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-top 16965  df-topon 16968  df-cn 17293  df-cnp 17294
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