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Theorem cnsscnp 20993
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 = 𝐽
Assertion
Ref Expression
cnsscnp (𝑃𝑋 → (𝐽 Cn 𝐾) ⊆ ((𝐽 CnP 𝐾)‘𝑃))

Proof of Theorem cnsscnp
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cnsscnp.1 . . . 4 𝑋 = 𝐽
21cncnpi 20992 . . 3 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑃𝑋) → 𝑓 ∈ ((𝐽 CnP 𝐾)‘𝑃))
32expcom 451 . 2 (𝑃𝑋 → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ ((𝐽 CnP 𝐾)‘𝑃)))
43ssrdv 3589 1 (𝑃𝑋 → (𝐽 Cn 𝐾) ⊆ ((𝐽 CnP 𝐾)‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wss 3555   cuni 4402  cfv 5847  (class class class)co 6604   Cn ccn 20938   CnP ccnp 20939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-top 20621  df-topon 20623  df-cn 20941  df-cnp 20942
This theorem is referenced by: (None)
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