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Theorem cnfex 41493
 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 2824 . . . . 5 𝐽 = 𝐽
21jctr 528 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
3 istopon 21506 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
42, 3sylibr 237 . . 3 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2824 . . . . 5 𝐾 = 𝐾
65jctr 528 . . . 4 (𝐾 ∈ Top → (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
7 istopon 21506 . . . 4 (𝐾 ∈ (TopOn‘ 𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
86, 7sylibr 237 . . 3 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
9 cnfval 21827 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ ( 𝐾m 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
104, 8, 9syl2an 598 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ ( 𝐾m 𝐽) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
11 uniexg 7449 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ V)
12 uniexg 7449 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ V)
13 mapvalg 8399 . . . . 5 (( 𝐾 ∈ V ∧ 𝐽 ∈ V) → ( 𝐾m 𝐽) = {𝑓𝑓: 𝐽 𝐾})
1411, 12, 13syl2anr 599 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾m 𝐽) = {𝑓𝑓: 𝐽 𝐾})
15 mapex 8395 . . . . 5 (( 𝐽 ∈ V ∧ 𝐾 ∈ V) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1612, 11, 15syl2an 598 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1714, 16eqeltrd 2916 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾m 𝐽) ∈ V)
18 rabexg 5215 . . 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 2916 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {cab 2802  ∀wral 3132  {crab 3136  Vcvv 3479  ∪ cuni 4819  ◡ccnv 5535   “ cima 5539  ⟶wf 6332  ‘cfv 6336  (class class class)co 7138   ↑m cmap 8389  Topctop 21487  TopOnctopon 21504   Cn ccn 21818 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-map 8391  df-topon 21505  df-cn 21821 This theorem is referenced by:  stoweidlem53  42537  stoweidlem57  42541
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