Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnfex Structured version   Visualization version   GIF version

Theorem cnfex 38210
 Description: The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Assertion
Ref Expression
cnfex ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Proof of Theorem cnfex
Dummy variables 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 𝐽 = 𝐽
21jctr 563 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
3 istopon 20540 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
42, 3sylibr 223 . . 3 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2610 . . . . 5 𝐾 = 𝐾
65jctr 563 . . . 4 (𝐾 ∈ Top → (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
7 istopon 20540 . . . 4 (𝐾 ∈ (TopOn‘ 𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
86, 7sylibr 223 . . 3 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
9 cnfval 20847 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
104, 8, 9syl2an 493 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
11 uniexg 6853 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ V)
12 uniexg 6853 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ V)
13 mapvalg 7754 . . . . 5 (( 𝐾 ∈ V ∧ 𝐽 ∈ V) → ( 𝐾𝑚 𝐽) = {𝑓𝑓: 𝐽 𝐾})
1411, 12, 13syl2anr 494 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾𝑚 𝐽) = {𝑓𝑓: 𝐽 𝐾})
15 mapex 7750 . . . . 5 (( 𝐽 ∈ V ∧ 𝐾 ∈ V) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1612, 11, 15syl2an 493 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1714, 16eqeltrd 2688 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾𝑚 𝐽) ∈ V)
18 rabexg 4739 . . 3 (( 𝐾𝑚 𝐽) ∈ V → {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ∈ V)
1917, 18syl 17 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ ( 𝐾𝑚 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ∈ V)
2010, 19eqeltrd 2688 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173  ∪ cuni 4372  ◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Topctop 20517  TopOnctopon 20518   Cn ccn 20838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-topon 20523  df-cn 20841 This theorem is referenced by:  stoweidlem53  38946  stoweidlem57  38950  stoweidlem59  38952
 Copyright terms: Public domain W3C validator