Theorem cnovex 15220

Description: The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)

Assertion

Ref	Expression
cnovex	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Proof of Theorem cnovex

Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		toptopon2 15043	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2		toptopon2 15043	. . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
3		cnfval 15218	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
4	1, 2, 3	syl2anb 291	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
5		uniexg 4580	. . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
6		uniexg 4580	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
7		mapvalg 6922	. . . . 5 ⊢ ((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) = {𝑧 ∣ 𝑧:∪ 𝐽⟶∪ 𝐾})
8	5, 6, 7	syl2anr 290	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) = {𝑧 ∣ 𝑧:∪ 𝐽⟶∪ 𝐾})
9		mapex 6918	. . . . 5 ⊢ ((∪ 𝐽 ∈ V ∧ ∪ 𝐾 ∈ V) → {𝑧 ∣ 𝑧:∪ 𝐽⟶∪ 𝐾} ∈ V)
10	6, 5, 9	syl2an 289	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑧 ∣ 𝑧:∪ 𝐽⟶∪ 𝐾} ∈ V)
11	8, 10	eqeltrd 2315	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∈ V)
12		rabexg 4274	. . 3 ⊢ ((∪ 𝐾 ↑𝑚 ∪ 𝐽) ∈ V → {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
13	11, 12	syl 14	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
14	4, 13	eqeltrd 2315	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  {crab 2532  Vcvv 2821  ∪ cuni 3930  ^◡ccnv 4768   “ cima 4772  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075   ↑_𝑚 cmap 6912  Topctop 15021  TopOnctopon 15034   Cn ccn 15209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-top 15022  df-topon 15035  df-cn 15212

This theorem is referenced by:  hmeofn  15326  hmeofvalg  15327