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Theorem cnfex 41162
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 2818 . . . . 5 𝐽 = 𝐽
21jctr 525 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
3 istopon 21448 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
42, 3sylibr 235 . . 3 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2818 . . . . 5 𝐾 = 𝐾
65jctr 525 . . . 4 (𝐾 ∈ Top → (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
7 istopon 21448 . . . 4 (𝐾 ∈ (TopOn‘ 𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
86, 7sylibr 235 . . 3 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
9 cnfval 21769 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ ( 𝐾m 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
104, 8, 9syl2an 595 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ ( 𝐾m 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
11 uniexg 7456 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ V)
12 uniexg 7456 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ V)
13 mapvalg 8405 . . . . 5 (( 𝐾 ∈ V ∧ 𝐽 ∈ V) → ( 𝐾m 𝐽) = {𝑓𝑓: 𝐽 𝐾})
1411, 12, 13syl2anr 596 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾m 𝐽) = {𝑓𝑓: 𝐽 𝐾})
15 mapex 8401 . . . . 5 (( 𝐽 ∈ V ∧ 𝐾 ∈ V) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1612, 11, 15syl2an 595 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1714, 16eqeltrd 2910 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾m 𝐽) ∈ V)
18 rabexg 5225 . . 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 2910 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wral 3135  {crab 3139  Vcvv 3492   cuni 4830  ccnv 5547  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  Topctop 21429  TopOnctopon 21446   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-topon 21447  df-cn 21763
This theorem is referenced by:  stoweidlem53  42215  stoweidlem57  42219
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