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Mirrors  >  Home  >  MPE Home  >  Th. List  >  xkoval Structured version   Visualization version   GIF version
Theorem xkoval 23091
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 486 . . . . . . . . . . . . 13 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ π‘Ÿ = 𝑅)
21unieqd 4923 . . . . . . . . . . . 12 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ βˆͺ π‘Ÿ = βˆͺ 𝑅)
3 xkoval.x . . . . . . . . . . . 12 𝑋 = βˆͺ 𝑅
42, 3eqtr4di 2791 . . . . . . . . . . 11 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ βˆͺ π‘Ÿ = 𝑋)
54pweqd 4620 . . . . . . . . . 10 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ 𝒫 βˆͺ π‘Ÿ = 𝒫 𝑋)
61oveq1d 7424 . . . . . . . . . . 11 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ β†Ύt π‘₯) = (𝑅 β†Ύt π‘₯))
76eleq1d 2819 . . . . . . . . . 10 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ ((π‘Ÿ β†Ύt π‘₯) ∈ Comp ↔ (𝑅 β†Ύt π‘₯) ∈ Comp))
85, 7rabeqbidv 3450 . . . . . . . . 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 2791 . . . . . . . 8 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ {π‘₯ ∈ 𝒫 βˆͺ π‘Ÿ ∣ (π‘Ÿ β†Ύt π‘₯) ∈ Comp} = 𝐾)
11 simpl 484 . . . . . . . 8 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ 𝑠 = 𝑆)
121, 11oveq12d 7427 . . . . . . . . 9 ((𝑠 = 𝑆 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ Cn 𝑠) = (𝑅 Cn 𝑆))
1312rabeqdv 3448 . . . . . . . 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 7484 . . . . . . 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 2791 . . . . . 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 5938 . . . . 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 6896 . . . 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 6896 . . 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 23067 . . 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 6905 . . 3 (topGenβ€˜(fiβ€˜ran 𝑇)) ∈ V
2219, 20, 21ovmpoa 7563 . 2 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) β†’ (𝑆 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran 𝑇)))
2322ancoms 460 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  ran crn 5678   β€œ cima 5680  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  ficfi 9405   β†Ύt crest 17366  topGenctg 17383  Topctop 22395   Cn ccn 22728  Compccmp 22890   ↑ko cxko 23065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-xko 23067
This theorem is referenced by:  xkotop  23092  xkoopn  23093  xkouni  23103  xkoccn  23123  xkopt  23159  xkoco1cn  23161  xkoco2cn  23162  xkococn  23164  xkoinjcn  23191
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