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Theorem kgenval 23459
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 23458 . 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 4923 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
3 toponuni 22836 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
43eqcomd 2734 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 = 𝑋)
52, 4sylan9eqr 2790 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ βˆͺ 𝑗 = 𝑋)
65pweqd 4623 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ 𝒫 βˆͺ 𝑗 = 𝒫 𝑋)
7 oveq1 7433 . . . . . . 7 (𝑗 = 𝐽 β†’ (𝑗 β†Ύt π‘˜) = (𝐽 β†Ύt π‘˜))
87eleq1d 2814 . . . . . 6 (𝑗 = 𝐽 β†’ ((𝑗 β†Ύt π‘˜) ∈ Comp ↔ (𝐽 β†Ύt π‘˜) ∈ Comp))
97eleq2d 2815 . . . . . 6 (𝑗 = 𝐽 β†’ ((π‘₯ ∩ π‘˜) ∈ (𝑗 β†Ύt π‘˜) ↔ (π‘₯ ∩ π‘˜) ∈ (𝐽 β†Ύt π‘˜)))
108, 9imbi12d 343 . . . . 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 480 . . . 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 3340 . . 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 3448 . 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 22835 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
15 toponmax 22848 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
16 pwexg 5382 . . 3 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
17 rabexg 5337 . . 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 7018 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 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430  Vcvv 3473   ∩ cin 3948  π’« cpw 4606  βˆͺ cuni 4912  β€˜cfv 6553  (class class class)co 7426   β†Ύt crest 17409  Topctop 22815  TopOnctopon 22832  Compccmp 23310  π‘˜Genckgen 23457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-top 22816  df-topon 22833  df-kgen 23458
This theorem is referenced by:  elkgen  23460  kgentopon  23462
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