Theorem xkoval 23502

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

Hypotheses

Ref	Expression
xkoval.x	⊢ 𝑋 = ∪ 𝑅
xkoval.k	⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
xkoval.t	⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})

Assertion

Ref	Expression
xkoval	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran 𝑇)))

Distinct variable groups:   𝑣,𝑘,𝐾   𝑓,𝑘,𝑣,𝑥,𝑅   𝑆,𝑓,𝑘,𝑣,𝑥   𝑘,𝑋,𝑥

Allowed substitution hints:   𝑇(𝑥,𝑣,𝑓,𝑘)   𝐾(𝑥,𝑓)   𝑋(𝑣,𝑓)

Proof of Theorem xkoval

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
2	1	unieqd 4869	. . . . . . . . . . . 12 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = ∪ 𝑅)
3		xkoval.x	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝑅
4	2, 3	eqtr4di 2784	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = 𝑋)
5	4	pweqd 4564	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝒫 ∪ 𝑟 = 𝒫 𝑋)
6	1	oveq1d 7361	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 ↾t 𝑥) = (𝑅 ↾t 𝑥))
7	6	eleq1d 2816	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑟 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑥) ∈ Comp))
8	5, 7	rabeqbidv 3413	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp})
9		xkoval.k	. . . . . . . . 9 ⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
10	8, 9	eqtr4di 2784	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp} = 𝐾)
11		simpl 482	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑠 = 𝑆)
12	1, 11	oveq12d 7364	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
13	12	rabeqdv 3410	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
14	10, 11, 13	mpoeq123dv 7421	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
15		xkoval.t	. . . . . . 7 ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
16	14, 15	eqtr4di 2784	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = 𝑇)
17	16	rneqd 5877	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = ran 𝑇)
18	17	fveq2d 6826	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) = (fi‘ran 𝑇))
19	18	fveq2d 6826	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) = (topGen‘(fi‘ran 𝑇)))
20		df-xko 23478	. . 3 ⊢ ↑ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
21		fvex 6835	. . 3 ⊢ (topGen‘(fi‘ran 𝑇)) ∈ V
22	19, 20, 21	ovmpoa 7501	. 2 ⊢ ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
23	22	ancoms 458	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran 𝑇)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856  ran crn 5615   “ cima 5617  ‘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-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-xko 23478

This theorem is referenced by:  xkotop  23503  xkoopn  23504  xkouni  23514  xkoccn  23534  xkopt  23570  xkoco1cn  23572  xkoco2cn  23573  xkococn  23575  xkoinjcn  23602