Theorem cnfex 38210

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 2610	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	jctr 563	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
3		istopon 20540	. . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
4	2, 3	sylibr 223	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2610	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
6	5	jctr 563	. . . 4 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
7		istopon 20540	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) ↔ (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
8	6, 7	sylibr 223	. . 3 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
9		cnfval 20847	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
10	4, 8, 9	syl2an 493	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
11		uniexg 6853	. . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
12		uniexg 6853	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
13		mapvalg 7754	. . . . 5 ⊢ ((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
14	11, 12, 13	syl2anr 494	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
15		mapex 7750	. . . . 5 ⊢ ((∪ 𝐽 ∈ V ∧ ∪ 𝐾 ∈ V) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
16	12, 11, 15	syl2an 493	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
17	14, 16	eqeltrd 2688	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑𝑚 ∪ 𝐽) ∈ V)
18		rabexg 4739	. . 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 2688	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173  ∪ cuni 4372  ^◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Topctop 20517  TopOnctopon 20518   Cn ccn 20838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-topon 20523  df-cn 20841

This theorem is referenced by:  stoweidlem53  38946  stoweidlem57  38950  stoweidlem59  38952