Theorem xkouni 23918

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 6074	. . . . . . . . 9 ⊢ (𝑓 “ ∅) = ∅
2		0ss 4349	. . . . . . . . 9 ⊢ ∅ ⊆ ∪ 𝑆
3	1, 2	eqsstri 3954	. . . . . . . 8 ⊢ (𝑓 “ ∅) ⊆ ∪ 𝑆
4	3	a1i 11	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (𝑓 “ ∅) ⊆ ∪ 𝑆)
5	4	ralrimiva 3154	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ ∪ 𝑆)
6		rabid2 3444	. . . . . 6 ⊢ ((𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆} ↔ ∀𝑓 ∈ (𝑅 Cn 𝑆)(𝑓 “ ∅) ⊆ ∪ 𝑆)
7	5, 6	sylibr 237	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆})
8		eqid 2760	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
9		simpl 488	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
10		simpr 490	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
11		0ss 4349	. . . . . . 7 ⊢ ∅ ⊆ ∪ 𝑅
12	11	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∅ ⊆ ∪ 𝑅)
13		rest0 23487	. . . . . . . 8 ⊢ (𝑅 ∈ Top → (𝑅 ↾t ∅) = {∅})
14	13	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) = {∅})
15		0cmp 23712	. . . . . . 7 ⊢ {∅} ∈ Comp
16	14, 15	eqeltrdi 2868	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ↾t ∅) ∈ Comp)
17		eqid 2760	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
18	17	topopn 23224	. . . . . . 7 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
19	18	adantl 487	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝑆 ∈ 𝑆)
20	8, 9, 10, 12, 16, 19	xkoopn 23908	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ ∅) ⊆ ∪ 𝑆} ∈ (𝑆 ↑ko 𝑅))
21	7, 20	eqeltrd 2860	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
22		xkouni.1	. . . 4 ⊢ 𝐽 = (𝑆 ↑ko 𝑅)
23	21, 22	eleqtrrdi 2871	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ 𝐽)
24		elssuni 4898	. . 3 ⊢ ((𝑅 Cn 𝑆) ∈ 𝐽 → (𝑅 Cn 𝑆) ⊆ ∪ 𝐽)
25	23, 24	syl 18	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ ∪ 𝐽)
26		eqid 2760	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
27		eqid 2760	. . . . . 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 23906	. . . . 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 4879	. . . 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 4878	. . . 4 ⊢ ∪ 𝐽 = ∪ (𝑆 ↑ko 𝑅)
31		ovex 7453	. . . . . . . 8 ⊢ (𝑅 Cn 𝑆) ∈ V
32	31	pwex 5341	. . . . . . 7 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
33	8, 26, 27	xkotf 23904	. . . . . . . 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 6717	. . . . . . . 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 5283	. . . . . 6 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
37		fiuni 9420	. . . . . 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 6898	. . . . . 6 ⊢ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ V
40		unitg 23285	. . . . . 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 2786	. . . 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 2820	. . 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 5059	. . . 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 221	. . 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 3942	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝐽 ⊆ (𝑅 Cn 𝑆))
48	25, 47	eqssd 3925	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866   × cxp 5648  ran crn 5651   “ cima 5653  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  ficfi 9402   ↾_t crest 17591  topGenctg 17608  Topctop 23211   Cn ccn 23542  Compccmp 23704   ↑_ko cxko 23880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-1o 8476  df-2o 8477  df-en 8974  df-fin 8977  df-fi 9403  df-rest 17593  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-cmp 23705  df-xko 23882

This theorem is used by:  xkotopon  23919  xkohaus  23972  xkoptsub  23973