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Theorem mrisval 17574
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 6925 . . . . 5 (Mooreβ€˜π‘‹) βŠ† βˆͺ ran Moore
32sseli 3979 . . . 4 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝐴 ∈ βˆͺ ran Moore)
4 unieq 4920 . . . . . . 7 (𝑐 = 𝐴 β†’ βˆͺ 𝑐 = βˆͺ 𝐴)
54pweqd 4620 . . . . . 6 (𝑐 = 𝐴 β†’ 𝒫 βˆͺ 𝑐 = 𝒫 βˆͺ 𝐴)
6 fveq2 6892 . . . . . . . . . . 11 (𝑐 = 𝐴 β†’ (mrClsβ€˜π‘) = (mrClsβ€˜π΄))
7 mrisval.1 . . . . . . . . . . 11 𝑁 = (mrClsβ€˜π΄)
86, 7eqtr4di 2791 . . . . . . . . . 10 (𝑐 = 𝐴 β†’ (mrClsβ€˜π‘) = 𝑁)
98fveq1d 6894 . . . . . . . . 9 (𝑐 = 𝐴 β†’ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯})) = (π‘β€˜(𝑠 βˆ– {π‘₯})))
109eleq2d 2820 . . . . . . . 8 (𝑐 = 𝐴 β†’ (π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯})) ↔ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))))
1110notbid 318 . . . . . . 7 (𝑐 = 𝐴 β†’ (Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯})) ↔ Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))))
1211ralbidv 3178 . . . . . 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 3450 . . . . 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 17532 . . . . 5 mrInd = (𝑐 ∈ βˆͺ ran Moore ↦ {𝑠 ∈ 𝒫 βˆͺ 𝑐 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯}))})
15 vuniex 7729 . . . . . . 7 βˆͺ 𝑐 ∈ V
1615pwex 5379 . . . . . 6 𝒫 βˆͺ 𝑐 ∈ V
1716rabex 5333 . . . . 5 {𝑠 ∈ 𝒫 βˆͺ 𝑐 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ ((mrClsβ€˜π‘)β€˜(𝑠 βˆ– {π‘₯}))} ∈ V
1813, 14, 17fvmpt3i 7004 . . . 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 2785 . 2 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝐼 = {𝑠 ∈ 𝒫 βˆͺ 𝐴 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
21 mreuni 17544 . . . 4 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ βˆͺ 𝐴 = 𝑋)
2221pweqd 4620 . . 3 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝒫 βˆͺ 𝐴 = 𝒫 𝑋)
2322rabeqdv 3448 . 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 2773 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 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433   βˆ– cdif 3946  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909  ran crn 5678  β€˜cfv 6544  Moorecmre 17526  mrClscmrc 17527  mrIndcmri 17528
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-pow 5364  ax-pr 5428  ax-un 7725
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-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-mpt 5233  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-mre 17530  df-mri 17532
This theorem is referenced by:  ismri  17575
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