Theorem cnfex 45340

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.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	jctr 524	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
3		istopon 22860	. . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
4	2, 3	sylibr 234	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2737	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
6	5	jctr 524	. . . 4 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
7		istopon 22860	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) ↔ (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
8	6, 7	sylibr 234	. . 3 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
9		cnfval 23181	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
10	4, 8, 9	syl2an 597	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
11		uniexg 7687	. . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
12		uniexg 7687	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
13		mapvalg 8777	. . . . 5 ⊢ ((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) → (∪ 𝐾 ↑m ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
14	11, 12, 13	syl2anr 598	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑m ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
15		mapex 7885	. . . . 5 ⊢ ((∪ 𝐽 ∈ V ∧ ∪ 𝐾 ∈ V) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
16	12, 11, 15	syl2an 597	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
17	14, 16	eqeltrd 2837	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑m ∪ 𝐽) ∈ V)
18		rabexg 5283	. . 3 ⊢ ((∪ 𝐾 ↑m ∪ 𝐽) ∈ V → {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
19	17, 18	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
20	10, 19	eqeltrd 2837	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3400  Vcvv 3441  ∪ cuni 4864  ^◡ccnv 5624   “ cima 5628  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ↑_m cmap 8767  Topctop 22841  TopOnctopon 22858   Cn ccn 23172

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-topon 22859  df-cn 23175

This theorem is referenced by:  stoweidlem53  46364  stoweidlem57  46368