Theorem xkouni 23541

Description: The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
xkouni.1	⊢ 𝐽 = (𝑆 ↑ko 𝑅)

Assertion

Ref	Expression
xkouni	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽)

Proof of Theorem xkouni

Dummy variables 𝑓 𝑘 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ima0 6034	. . . . . . . . 9 ⊢ (𝑓 “ ∅) = ∅
2		0ss 4350	. . . . . . . . 9 ⊢ ∅ ⊆ ∪ 𝑆
3	1, 2	eqsstri 3978	. . . . . . . 8 ⊢ (𝑓 “ ∅) ⊆ ∪ 𝑆
4	3	a1i 11	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (𝑓 “ ∅) ⊆ ∪ 𝑆)
5	4	ralrimiva 3126	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ ∪ 𝑆)
6		rabid2 3430	. . . . . 6 ⊢ ((𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆} ↔ ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ ∪ 𝑆)
7	5, 6	sylibr 234	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆})
8		eqid 2734	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
9		simpl 482	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
10		simpr 484	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
11		0ss 4350	. . . . . . 7 ⊢ ∅ ⊆ ∪ 𝑅
12	11	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∅ ⊆ ∪ 𝑅)
13		rest0 23111	. . . . . . . 8 ⊢ (𝑅 ∈ Top → (𝑅 ↾t ∅) = {∅})
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) = {∅})
15		0cmp 23336	. . . . . . 7 ⊢ {∅} ∈ Comp
16	14, 15	eqeltrdi 2842	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) ∈ Comp)
17		eqid 2734	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
18	17	topopn 22848	. . . . . . 7 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
19	18	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝑆 ∈ 𝑆)
20	8, 9, 10, 12, 16, 19	xkoopn 23531	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆} ∈ (𝑆 ↑ko 𝑅))
21	7, 20	eqeltrd 2834	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
22		xkouni.1	. . . 4 ⊢ 𝐽 = (𝑆 ↑ko 𝑅)
23	21, 22	eleqtrrdi 2845	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ 𝐽)
24		elssuni 4892	. . 3 ⊢ ((𝑅 Cn 𝑆) ∈ 𝐽 → (𝑅 Cn 𝑆) ⊆ ∪ 𝐽)
25	23, 24	syl 17	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ ∪ 𝐽)
26		eqid 2734	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
27		eqid 2734	. . . . . 6 ⊢ (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
28	8, 26, 27	xkoval 23529	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
29	28	unieqd 4874	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ (𝑆 ↑ko 𝑅) = ∪ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
30	22	unieqi 4873	. . . 4 ⊢ ∪ 𝐽 = ∪ (𝑆 ↑ko 𝑅)
31		ovex 7389	. . . . . . . 8 ⊢ (𝑅 Cn 𝑆) ∈ V
32	31	pwex 5323	. . . . . . 7 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
33	8, 26, 27	xkotf 23527	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
34		frn 6667	. . . . . . . 8 ⊢ ((𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆)
36	32, 35	ssexi 5265	. . . . . 6 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
37		fiuni 9329	. . . . . 6 ⊢ (ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V → ∪ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = ∪ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))
38	36, 37	ax-mp 5	. . . . 5 ⊢ ∪ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = ∪ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
39		fvex 6845	. . . . . 6 ⊢ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ V
40		unitg 22909	. . . . . 6 ⊢ ((fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ V → ∪ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) = ∪ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))
41	39, 40	ax-mp 5	. . . . 5 ⊢ ∪ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) = ∪ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
42	38, 41	eqtr4i 2760	. . . 4 ⊢ ∪ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = ∪ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))
43	29, 30, 42	3eqtr4g 2794	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝐽 = ∪ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
44	35	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
45		sspwuni 5053	. . . 4 ⊢ (ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆) ↔ ∪ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ (𝑅 Cn 𝑆))
46	44, 45	sylib 218	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ (𝑅 Cn 𝑆))
47	43, 46	eqsstrd 3966	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝐽 ⊆ (𝑅 Cn 𝑆))
48	25, 47	eqssd 3949	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  {crab 3397  Vcvv 3438   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  {csn 4578  ∪ cuni 4861   × cxp 5620  ran crn 5623   “ cima 5625  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  ficfi 9311   ↾_t crest 17338  topGenctg 17355  Topctop 22835   Cn ccn 23166  Compccmp 23328   ↑_ko cxko 23503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-1o 8395  df-2o 8396  df-en 8882  df-fin 8885  df-fi 9312  df-rest 17340  df-topgen 17361  df-top 22836  df-topon 22853  df-bases 22888  df-cmp 23329  df-xko 23505

This theorem is referenced by:  xkotopon  23542  xkohaus  23595  xkoptsub  23596