Theorem cnfex 46248

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 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	jctr 534	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
3		istopon 23257	. . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽))
4	2, 3	sylibr 237	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2763	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
6	5	jctr 534	. . . 4 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
7		istopon 23257	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) ↔ (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾))
8	6, 7	sylibr 237	. . 3 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
9		cnfval 23578	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
10	4, 8, 9	syl2an 608	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
11		uniexg 7778	. . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
12		uniexg 7778	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
13		mapvalg 8881	. . . . 5 ⊢ ((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) → (∪ 𝐾 ↑m ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
14	11, 12, 13	syl2anr 609	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑m ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾})
15		mapex 7973	. . . . 5 ⊢ ((∪ 𝐽 ∈ V ∧ ∪ 𝐾 ∈ V) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
16	12, 11, 15	syl2an 608	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V)
17	14, 16	eqeltrd 2864	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑m ∪ 𝐽) ∈ V)
18		rabexg 5303	. . 3 ⊢ ((∪ 𝐾 ↑m ∪ 𝐽) ∈ V → {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
19	17, 18	syl 18	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
20	10, 19	eqeltrd 2864	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  {crab 3416  Vcvv 3454  ∪ cuni 4870  ^◡ccnv 5654   “ cima 5658  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  Topctop 23238  TopOnctopon 23255   Cn ccn 23569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874  df-topon 23256  df-cn 23572

This theorem is used by:  stoweidlem53  47266  stoweidlem57  47270