Theorem cnfex 40005

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 2825	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	jctr 522	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
3		istopon 21087	. . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
4	2, 3	sylibr 226	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2825	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
6	5	jctr 522	. . . 4 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
7		istopon 21087	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) ↔ (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
8	6, 7	sylibr 226	. . 3 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
9		cnfval 21408	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
10	4, 8, 9	syl2an 591	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
11		uniexg 7215	. . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
12		uniexg 7215	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
13		mapvalg 8132	. . . . 5 ⊢ ((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
14	11, 12, 13	syl2anr 592	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
15		mapex 8128	. . . . 5 ⊢ ((∪ 𝐽 ∈ V ∧ ∪ 𝐾 ∈ V) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
16	12, 11, 15	syl2an 591	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
17	14, 16	eqeltrd 2906	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∈ V)
18		rabexg 5036	. . 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 2906	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {cab 2811  ∀wral 3117  {crab 3121  Vcvv 3414  ∪ cuni 4658  ^◡ccnv 5341   “ cima 5345  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905   ↑_𝑚 cmap 8122  Topctop 21068  TopOnctopon 21085   Cn ccn 21399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-topon 21086  df-cn 21402

This theorem is referenced by:  stoweidlem53  41064  stoweidlem57  41068