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Theorem cnfex 44178
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 2731 . . . . 5 𝐽 = 𝐽
21jctr 524 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
3 istopon 22735 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐽))
42, 3sylibr 233 . . 3 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2731 . . . . 5 𝐾 = 𝐾
65jctr 524 . . . 4 (𝐾 ∈ Top → (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
7 istopon 22735 . . . 4 (𝐾 ∈ (TopOn‘ 𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝐾))
86, 7sylibr 233 . . 3 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
9 cnfval 23058 . . 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 7734 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ V)
12 uniexg 7734 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ V)
13 mapvalg 8836 . . . . 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 8832 . . . . 5 (( 𝐽 ∈ V ∧ 𝐾 ∈ V) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1612, 11, 15syl2an 595 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓𝑓: 𝐽 𝐾} ∈ V)
1714, 16eqeltrd 2832 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ( 𝐾m 𝐽) ∈ V)
18 rabexg 5331 . . 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 2832 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  {cab 2708  wral 3060  {crab 3431  Vcvv 3473   cuni 4908  ccnv 5675  cima 5679  wf 6539  cfv 6543  (class class class)co 7412  m cmap 8826  Topctop 22716  TopOnctopon 22733   Cn ccn 23049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-topon 22734  df-cn 23052
This theorem is referenced by:  stoweidlem53  45231  stoweidlem57  45235
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