Theorem cnovex 15007

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 14830	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2		toptopon2 14830	. . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
3		cnfval 15005	. . 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 4542	. . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
6		uniexg 4542	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
7		mapvalg 6870	. . . . 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 6866	. . . . 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 2308	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∈ V)
12		rabexg 4238	. . 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 2308	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  {crab 2515  Vcvv 2803  ∪ cuni 3898  ^◡ccnv 4730   “ cima 4734  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ↑_𝑚 cmap 6860  Topctop 14808  TopOnctopon 14821   Cn ccn 14996

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-top 14809  df-topon 14822  df-cn 14999

This theorem is referenced by:  hmeofn  15113  hmeofvalg  15114