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

Theorem cnfex 39501
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 2651 . . . . 5 𝐽 = 𝐽
21jctr 564 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
3 istopon 20765 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
42, 3sylibr 224 . . 3 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2651 . . . . 5 𝐾 = 𝐾
65jctr 564 . . . 4 (𝐾 ∈ Top → (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
7 istopon 20765 . . . 4 (𝐾 ∈ (TopOn‘ 𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
86, 7sylibr 224 . . 3 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
9 cnfval 21085 . . 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 6997 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ V)
12 uniexg 6997 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ V)
13 mapvalg 7909 . . . . 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 7905 . . . . 5 (( 𝐽 ∈ V ∧ 𝐾 ∈ V) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1612, 11, 15syl2an 493 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1714, 16eqeltrd 2730 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾𝑚 𝐽) ∈ V)
18 rabexg 4844 . . 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 2730 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  {crab 2945  Vcvv 3231   cuni 4468  ccnv 5142  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Topctop 20746  TopOnctopon 20763   Cn ccn 21076
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-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-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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-topon 20764  df-cn 21079
This theorem is referenced by:  stoweidlem53  40588  stoweidlem57  40592
  Copyright terms: Public domain W3C validator