Theorem xkouni 22207

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 5945	. . . . . . . . 9 ⊢ (𝑓 “ ∅) = ∅
2		0ss 4350	. . . . . . . . 9 ⊢ ∅ ⊆ ∪ 𝑆
3	1, 2	eqsstri 4001	. . . . . . . 8 ⊢ (𝑓 “ ∅) ⊆ ∪ 𝑆
4	3	a1i 11	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (𝑓 “ ∅) ⊆ ∪ 𝑆)
5	4	ralrimiva 3182	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ ∪ 𝑆)
6		rabid2 3381	. . . . . 6 ⊢ ((𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆} ↔ ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ ∪ 𝑆)
7	5, 6	sylibr 236	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆})
8		eqid 2821	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
9		simpl 485	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
10		simpr 487	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
11		0ss 4350	. . . . . . 7 ⊢ ∅ ⊆ ∪ 𝑅
12	11	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∅ ⊆ ∪ 𝑅)
13		rest0 21777	. . . . . . . 8 ⊢ (𝑅 ∈ Top → (𝑅 ↾t ∅) = {∅})
14	13	adantr 483	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) = {∅})
15		0cmp 22002	. . . . . . 7 ⊢ {∅} ∈ Comp
16	14, 15	eqeltrdi 2921	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) ∈ Comp)
17		eqid 2821	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
18	17	topopn 21514	. . . . . . 7 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
19	18	adantl 484	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝑆 ∈ 𝑆)
20	8, 9, 10, 12, 16, 19	xkoopn 22197	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆} ∈ (𝑆 ↑ko 𝑅))
21	7, 20	eqeltrd 2913	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
22		xkouni.1	. . . 4 ⊢ 𝐽 = (𝑆 ↑ko 𝑅)
23	21, 22	eleqtrrdi 2924	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ 𝐽)
24		elssuni 4868	. . 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 2821	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
27		eqid 2821	. . . . . 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 22195	. . . . 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 4852	. . . 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 4851	. . . 4 ⊢ ∪ 𝐽 = ∪ (𝑆 ↑ko 𝑅)
31		ovex 7189	. . . . . . . 8 ⊢ (𝑅 Cn 𝑆) ∈ V
32	31	pwex 5281	. . . . . . 7 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
33	8, 26, 27	xkotf 22193	. . . . . . . 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 6520	. . . . . . . 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 5226	. . . . . 6 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
37		fiuni 8892	. . . . . 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 6683	. . . . . 6 ⊢ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ V
40		unitg 21575	. . . . . 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 2847	. . . 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 2881	. . 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 5022	. . . 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 220	. . 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 4005	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝐽 ⊆ (𝑅 Cn 𝑆))
48	25, 47	eqssd 3984	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {crab 3142  Vcvv 3494   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  ∪ cuni 4838   × cxp 5553  ran crn 5556   “ cima 5558  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  ficfi 8874   ↾_t crest 16694  topGenctg 16711  Topctop 21501   Cn ccn 21832  Compccmp 21994   ↑_ko cxko 22169

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-fin 8513  df-fi 8875  df-rest 16696  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-xko 22171

This theorem is referenced by:  xkotopon  22208  xkohaus  22261  xkoptsub  22262