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Theorem kgenval 23030
Description: Value of the compact generator. (The "k" in π‘˜Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgenval (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘˜Genβ€˜π½) = {π‘₯ ∈ 𝒫 𝑋 ∣ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))})
Distinct variable groups:   π‘₯,π‘˜,𝐽   π‘˜,𝑋,π‘₯
Proof of Theorem kgenval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 df-kgen 23029 . 2 π‘˜Gen = (𝑗 ∈ Top ↦ {π‘₯ ∈ 𝒫 βˆͺ 𝑗 ∣ βˆ€π‘˜ ∈ 𝒫 βˆͺ 𝑗((𝑗 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝑗 β†Ύt π‘˜))})
2 unieq 4918 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
3 toponuni 22407 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
43eqcomd 2738 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 = 𝑋)
52, 4sylan9eqr 2794 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ βˆͺ 𝑗 = 𝑋)
65pweqd 4618 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ 𝒫 βˆͺ 𝑗 = 𝒫 𝑋)
7 oveq1 7412 . . . . . . 7 (𝑗 = 𝐽 β†’ (𝑗 β†Ύt π‘˜) = (𝐽 β†Ύt π‘˜))
87eleq1d 2818 . . . . . 6 (𝑗 = 𝐽 β†’ ((𝑗 β†Ύt π‘˜) ∈ Comp ↔ (𝐽 β†Ύt π‘˜) ∈ Comp))
97eleq2d 2819 . . . . . 6 (𝑗 = 𝐽 β†’ ((π‘₯ ∩ π‘˜) ∈ (𝑗 β†Ύt π‘˜) ↔ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜)))
108, 9imbi12d 344 . . . . 5 (𝑗 = 𝐽 β†’ (((𝑗 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝑗 β†Ύt π‘˜)) ↔ ((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))))
1110adantl 482 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (((𝑗 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝑗 β†Ύt π‘˜)) ↔ ((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))))
126, 11raleqbidv 3342 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (βˆ€π‘˜ ∈ 𝒫 βˆͺ 𝑗((𝑗 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝑗 β†Ύt π‘˜)) ↔ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))))
136, 12rabeqbidv 3449 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ {π‘₯ ∈ 𝒫 βˆͺ 𝑗 ∣ βˆ€π‘˜ ∈ 𝒫 βˆͺ 𝑗((𝑗 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝑗 β†Ύt π‘˜))} = {π‘₯ ∈ 𝒫 𝑋 ∣ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))})
14 topontop 22406 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
15 toponmax 22419 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
16 pwexg 5375 . . 3 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
17 rabexg 5330 . . 3 (𝒫 𝑋 ∈ V β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))} ∈ V)
1815, 16, 173syl 18 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))} ∈ V)
191, 13, 14, 18fvmptd2 7003 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘˜Genβ€˜π½) = {π‘₯ ∈ 𝒫 𝑋 ∣ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  Vcvv 3474   ∩ cin 3946  π’« cpw 4601  βˆͺ cuni 4907  β€˜cfv 6540  (class class class)co 7405   β†Ύt crest 17362  Topctop 22386  TopOnctopon 22403  Compccmp 22881  π‘˜Genckgen 23028
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-pow 5362  ax-pr 5426  ax-un 7721
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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  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-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-top 22387  df-topon 22404  df-kgen 23029
This theorem is referenced by:  elkgen  23031  kgentopon  23033
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