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Theorem mrisval 17578
Description: Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrisval.1 𝑁 = (mrClsβ€˜π΄)
mrisval.2 𝐼 = (mrIndβ€˜π΄)
Assertion
Ref Expression
mrisval (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
Distinct variable groups:   𝐴,𝑠,π‘₯   𝑋,𝑠
Allowed substitution hints:   𝐼(π‘₯,𝑠)   𝑁(π‘₯,𝑠)   𝑋(π‘₯)

Proof of Theorem mrisval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 mrisval.2 . . 3 𝐼 = (mrIndβ€˜π΄)
2 fvssunirn 6923 . . . . 5 (Mooreβ€˜π‘‹) βŠ† βˆͺ ran Moore
32sseli 3977 . . . 4 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝐴 ∈ βˆͺ ran Moore)
4 unieq 4918 . . . . . . 7 (𝑐 = 𝐴 β†’ βˆͺ 𝑐 = βˆͺ 𝐴)
54pweqd 4618 . . . . . 6 (𝑐 = 𝐴 β†’ 𝒫 βˆͺ 𝑐 = 𝒫 βˆͺ 𝐴)
6 fveq2 6890 . . . . . . . . . . 11 (𝑐 = 𝐴 β†’ (mrClsβ€˜π‘) = (mrClsβ€˜π΄))
7 mrisval.1 . . . . . . . . . . 11 𝑁 = (mrClsβ€˜π΄)
86, 7eqtr4di 2788 . . . . . . . . . 10 (𝑐 = 𝐴 β†’ (mrClsβ€˜π‘) = 𝑁)
98fveq1d 6892 . . . . . . . . 9 (𝑐 = 𝐴 β†’ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯})) = (π‘β€˜(𝑠 βˆ– {π‘₯})))
109eleq2d 2817 . . . . . . . 8 (𝑐 = 𝐴 β†’ (π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯})) ↔ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))))
1110notbid 317 . . . . . . 7 (𝑐 = 𝐴 β†’ (Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯})) ↔ Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))))
1211ralbidv 3175 . . . . . 6 (𝑐 = 𝐴 β†’ (βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯})) ↔ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))))
135, 12rabeqbidv 3447 . . . . 5 (𝑐 = 𝐴 β†’ {𝑠 ∈ 𝒫 βˆͺ 𝑐 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯}))} = {𝑠 ∈ 𝒫 βˆͺ 𝐴 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
14 df-mri 17536 . . . . 5 mrInd = (𝑐 ∈ βˆͺ ran Moore ↦ {𝑠 ∈ 𝒫 βˆͺ 𝑐 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯}))})
15 vuniex 7731 . . . . . . 7 βˆͺ 𝑐 ∈ V
1615pwex 5377 . . . . . 6 𝒫 βˆͺ 𝑐 ∈ V
1716rabex 5331 . . . . 5 {𝑠 ∈ 𝒫 βˆͺ 𝑐 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯}))} ∈ V
1813, 14, 17fvmpt3i 7002 . . . 4 (𝐴 ∈ βˆͺ ran Moore β†’ (mrIndβ€˜π΄) = {𝑠 ∈ 𝒫 βˆͺ 𝐴 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
193, 18syl 17 . . 3 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ (mrIndβ€˜π΄) = {𝑠 ∈ 𝒫 βˆͺ 𝐴 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
201, 19eqtrid 2782 . 2 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝐼 = {𝑠 ∈ 𝒫 βˆͺ 𝐴 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
21 mreuni 17548 . . . 4 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ βˆͺ 𝐴 = 𝑋)
2221pweqd 4618 . . 3 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝒫 βˆͺ 𝐴 = 𝒫 𝑋)
2322rabeqdv 3445 . 2 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ {𝑠 ∈ 𝒫 βˆͺ 𝐴 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))} = {𝑠 ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
2420, 23eqtrd 2770 1 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430   βˆ– cdif 3944  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907  ran crn 5676  β€˜cfv 6542  Moorecmre 17530  mrClscmrc 17531  mrIndcmri 17532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-mre 17534  df-mri 17536
This theorem is referenced by:  ismri  17579
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