Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnpcl Structured version   Visualization version   GIF version

Theorem cnpcl 21100
 Description: The value of a continuous function from 𝐽 to 𝐾 at point 𝑃 belongs to the underlying set of topology 𝐾. (Contributed by FL, 27-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
iscnp2.1 𝑋 = 𝐽
iscnp2.2 𝑌 = 𝐾
Assertion
Ref Expression
cnpcl ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)

Proof of Theorem cnpcl
StepHypRef Expression
1 iscnp2.1 . . 3 𝑋 = 𝐽
2 iscnp2.2 . . 3 𝑌 = 𝐾
31, 2cnpf 21099 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋𝑌)
43ffvelrnda 6399 1 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∪ cuni 4468  ‘cfv 5926  (class class class)co 6690   CnP ccnp 21077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-top 20747  df-topon 20764  df-cnp 21080 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator