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Mirrors  >  Home  >  MPE Home  >  Th. List  >  xkoval Structured version   Visualization version   GIF version
Theorem xkoval 23082
Description: Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoval.x 𝑋 = βˆͺ 𝑅
xkoval.k 𝐾 = {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}
xkoval.t 𝑇 = (π‘˜ ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})
Assertion
Ref Expression
xkoval ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran 𝑇)))
Distinct variable groups:   𝑣,π‘˜,𝐾   𝑓,π‘˜,𝑣,π‘₯,𝑅   𝑆,𝑓,π‘˜,𝑣,π‘₯   π‘˜,𝑋,π‘₯
Allowed substitution hints:   𝑇(π‘₯,𝑣,𝑓,π‘˜)   𝐾(π‘₯,𝑓)   𝑋(𝑣,𝑓)
Proof of Theorem xkoval
Dummy variables 𝑠 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . . . . . . . . . . 13 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ π‘Ÿ = 𝑅)
21unieqd 4921 . . . . . . . . . . . 12 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ βˆͺ π‘Ÿ = βˆͺ 𝑅)
3 xkoval.x . . . . . . . . . . . 12 𝑋 = βˆͺ 𝑅
42, 3eqtr4di 2790 . . . . . . . . . . 11 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ βˆͺ π‘Ÿ = 𝑋)
54pweqd 4618 . . . . . . . . . 10 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ 𝒫 βˆͺ π‘Ÿ = 𝒫 𝑋)
61oveq1d 7420 . . . . . . . . . . 11 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ β†Ύt π‘₯) = (𝑅 β†Ύt π‘₯))
76eleq1d 2818 . . . . . . . . . 10 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ ((π‘Ÿ β†Ύt π‘₯) ∈ Comp ↔ (𝑅 β†Ύt π‘₯) ∈ Comp))
85, 7rabeqbidv 3449 . . . . . . . . 9 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp} = {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp})
9 xkoval.k . . . . . . . . 9 𝐾 = {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}
108, 9eqtr4di 2790 . . . . . . . 8 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp} = 𝐾)
11 simpl 483 . . . . . . . 8 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ 𝑠 = 𝑆)
121, 11oveq12d 7423 . . . . . . . . 9 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ Cn 𝑠) = (𝑅 Cn 𝑆))
1312rabeqdv 3447 . . . . . . . 8 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ {𝑓 ∈ (π‘Ÿ Cn 𝑠) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})
1410, 11, 13mpoeq123dv 7480 . . . . . . 7 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (π‘˜ ∈ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (π‘Ÿ Cn 𝑠) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) = (π‘˜ ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))
15 xkoval.t . . . . . . 7 𝑇 = (π‘˜ ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})
1614, 15eqtr4di 2790 . . . . . 6 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (π‘˜ ∈ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (π‘Ÿ Cn 𝑠) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) = 𝑇)
1716rneqd 5935 . . . . 5 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (π‘Ÿ Cn 𝑠) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) = ran 𝑇)
1817fveq2d 6892 . . . 4 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (π‘Ÿ Cn 𝑠) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})) = (fiβ€˜ran 𝑇))
1918fveq2d 6892 . . 3 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (π‘Ÿ Cn 𝑠) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))) = (topGenβ€˜(fiβ€˜ran 𝑇)))
20 df-xko 23058 . . 3 ↑ko = (𝑠 ∈ Top, π‘Ÿ ∈ Top ↦ (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (π‘Ÿ Cn 𝑠) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))))
21 fvex 6901 . . 3 (topGenβ€˜(fiβ€˜ran 𝑇)) ∈ V
2219, 20, 21ovmpoa 7559 . 2 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) β†’ (𝑆 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran 𝑇)))
2322ancoms 459 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran 𝑇)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907  ran crn 5676   β€œ cima 5678  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  ficfi 9401   β†Ύt crest 17362  topGenctg 17379  Topctop 22386   Cn ccn 22719  Compccmp 22881   ↑ko cxko 23056
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-xko 23058
This theorem is referenced by:  xkotop  23083  xkoopn  23084  xkouni  23094  xkoccn  23114  xkopt  23150  xkoco1cn  23152  xkoco2cn  23153  xkococn  23155  xkoinjcn  23182
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