Theorem xkoval 23535

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 4877	. . . . . . . . . . . 12 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = ∪ 𝑅)
3		xkoval.x	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝑅
4	2, 3	eqtr4di 2790	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = 𝑋)
5	4	pweqd 4572	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝒫 ∪ 𝑟 = 𝒫 𝑋)
6	1	oveq1d 7375	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 ↾t 𝑥) = (𝑅 ↾t 𝑥))
7	6	eleq1d 2822	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑟 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑥) ∈ Comp))
8	5, 7	rabeqbidv 3418	. . . . . . . . 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 2790	. . . . . . . 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 7378	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆))
13	12	rabeqdv 3415	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
14	10, 11, 13	mpoeq123dv 7435	. . . . . . 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 2790	. . . . . 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 5888	. . . . 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 6839	. . . 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 6839	. . 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 23511	. . 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 6848	. . 3 ⊢ (topGen‘(fi‘ran 𝑇)) ∈ V
22	19, 20, 21	ovmpoa 7515	. 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 1542   ∈ wcel 2114  {crab 3400   ⊆ wss 3902  𝒫 cpw 4555  ∪ cuni 4864  ran crn 5626   “ cima 5628  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  ficfi 9317   ↾_t crest 17344  topGenctg 17361  Topctop 22841   Cn ccn 23172  Compccmp 23334   ↑_ko cxko 23509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-xko 23511

This theorem is referenced by:  xkotop  23536  xkoopn  23537  xkouni  23547  xkoccn  23567  xkopt  23603  xkoco1cn  23605  xkoco2cn  23606  xkococn  23608  xkoinjcn  23635