Theorem xkoval 23585

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 483	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
2	1	unieqd 4928	. . . . . . . . . . . 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 4624	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝒫 ∪ 𝑟 = 𝒫 𝑋)
6	1	oveq1d 7441	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 ↾t 𝑥) = (𝑅 ↾t 𝑥))
7	6	eleq1d 2811	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑟 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑥) ∈ Comp))
8	5, 7	rabeqbidv 3437	. . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑠 = 𝑆)
12	1, 11	oveq12d 7444	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
13	12	rabeqdv 3435	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
14	10, 11, 13	mpoeq123dv 7502	. . . . . . 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 5946	. . . . 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 6907	. . . 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 6907	. . 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 23561	. . 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 6916	. . 3 ⊢ (topGen‘(fi‘ran 𝑇)) ∈ V
22	19, 20, 21	ovmpoa 7583	. 2 ⊢ ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
23	22	ancoms 457	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran 𝑇)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  {crab 3419   ⊆ wss 3947  𝒫 cpw 4607  ∪ cuni 4915  ran crn 5685   “ cima 5687  ‘cfv 6556  (class class class)co 7426   ∈ cmpo 7428  ficfi 9455   ↾_t crest 17437  topGenctg 17454  Topctop 22889   Cn ccn 23222  Compccmp 23384   ↑_ko cxko 23559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6508  df-fun 6558  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-xko 23561

This theorem is referenced by:  xkotop  23586  xkoopn  23587  xkouni  23597  xkoccn  23617  xkopt  23653  xkoco1cn  23655  xkoco2cn  23656  xkococn  23658  xkoinjcn  23685