Theorem cnsscnp 23203

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	⊢ (𝑃 ∈ 𝑋 → (𝐽 Cn 𝐾) ⊆ ((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 = ∪ 𝐽
2	1	cncnpi 23202	. . 3 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑃 ∈ 𝑋) → 𝑓 ∈ ((𝐽 CnP 𝐾)‘𝑃))
3	2	expcom 412	. 2 ⊢ (𝑃 ∈ 𝑋 → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ ((𝐽 CnP 𝐾)‘𝑃)))
4	3	ssrdv 3988	1 ⊢ (𝑃 ∈ 𝑋 → (𝐽 Cn 𝐾) ⊆ ((𝐽 CnP 𝐾)‘𝑃))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ⊆ wss 3949  ∪ cuni 4912  ‘cfv 6553  (class class class)co 7426   Cn ccn 23148   CnP ccnp 23149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-top 22816  df-topon 22833  df-cn 23151  df-cnp 23152

This theorem is referenced by: (None)