Theorem cnsscnp 15225

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.

Step	Hyp	Ref	Expression
1		cnsscnp.1	. . . 4 |- X = U. J
2	1	cncnpi 15224	. . 3 |- ( ( f e. ( J Cn K ) /\ P e. X ) -> f e. ( ( J CnP K ) ` P ) )
3	2	expcom 116	. 2 |- ( P e. X -> ( f e. ( J Cn K ) -> f e. ( ( J CnP K ) ` P ) ) )
4	3	ssrdv 3248	1 |- ( P e. X -> ( J Cn K ) C_ ( ( J CnP K ) ` P ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   C_ wss 3214   U.cuni 3920   ` cfv 5358  (class class class)co 6059   Cn ccn 15181   CnP ccnp 15182

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-map 6898  df-top 14994  df-topon 15007  df-cn 15184  df-cnp 15185

This theorem is referenced by: (None)