Theorem xkouni 23514

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 6025	. . . . . . . . 9 ⊢ (𝑓 “ ∅) = ∅
2		0ss 4347	. . . . . . . . 9 ⊢ ∅ ⊆ ∪ 𝑆
3	1, 2	eqsstri 3976	. . . . . . . 8 ⊢ (𝑓 “ ∅) ⊆ ∪ 𝑆
4	3	a1i 11	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (𝑓 “ ∅) ⊆ ∪ 𝑆)
5	4	ralrimiva 3124	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ ∪ 𝑆)
6		rabid2 3428	. . . . . 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 2731	. . . . . 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 4347	. . . . . . 7 ⊢ ∅ ⊆ ∪ 𝑅
12	11	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∅ ⊆ ∪ 𝑅)
13		rest0 23084	. . . . . . . 8 ⊢ (𝑅 ∈ Top → (𝑅 ↾t ∅) = {∅})
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) = {∅})
15		0cmp 23309	. . . . . . 7 ⊢ {∅} ∈ Comp
16	14, 15	eqeltrdi 2839	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) ∈ Comp)
17		eqid 2731	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
18	17	topopn 22821	. . . . . . 7 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
19	18	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝑆 ∈ 𝑆)
20	8, 9, 10, 12, 16, 19	xkoopn 23504	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆} ∈ (𝑆 ↑ko 𝑅))
21	7, 20	eqeltrd 2831	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
22		xkouni.1	. . . 4 ⊢ 𝐽 = (𝑆 ↑ko 𝑅)
23	21, 22	eleqtrrdi 2842	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ 𝐽)
24		elssuni 4887	. . 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 2731	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
27		eqid 2731	. . . . . 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 23502	. . . . 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 4869	. . . 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 4868	. . . 4 ⊢ ∪ 𝐽 = ∪ (𝑆 ↑ko 𝑅)
31		ovex 7379	. . . . . . . 8 ⊢ (𝑅 Cn 𝑆) ∈ V
32	31	pwex 5316	. . . . . . 7 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
33	8, 26, 27	xkotf 23500	. . . . . . . 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 6658	. . . . . . . 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 5258	. . . . . 6 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
37		fiuni 9312	. . . . . 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 6835	. . . . . 6 ⊢ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ V
40		unitg 22882	. . . . . 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 2757	. . . 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 2791	. . 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 5046	. . . 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 3964	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝐽 ⊆ (𝑅 Cn 𝑆))
48	25, 47	eqssd 3947	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  {csn 4573  ∪ cuni 4856   × cxp 5612  ran crn 5615   “ cima 5617  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  ficfi 9294   ↾_t crest 17324  topGenctg 17341  Topctop 22808   Cn ccn 23139  Compccmp 23301   ↑_ko cxko 23476

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-1o 8385  df-2o 8386  df-en 8870  df-fin 8873  df-fi 9295  df-rest 17326  df-topgen 17347  df-top 22809  df-topon 22826  df-bases 22861  df-cmp 23302  df-xko 23478

This theorem is referenced by:  xkotopon  23515  xkohaus  23568  xkoptsub  23569