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Theorem ismri 16892
Description: Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri.1 𝑁 = (mrCls‘𝐴)
ismri.2 𝐼 = (mrInd‘𝐴)
Assertion
Ref Expression
ismri (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑆
Allowed substitution hints:   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem ismri
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ismri.1 . . . . 5 𝑁 = (mrCls‘𝐴)
2 ismri.2 . . . . 5 𝐼 = (mrInd‘𝐴)
31, 2mrisval 16891 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
43eleq2d 2903 . . 3 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}))
5 difeq1 4096 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
65fveq2d 6671 . . . . . . 7 (𝑠 = 𝑆 → (𝑁‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥})))
76eleq2d 2903 . . . . . 6 (𝑠 = 𝑆 → (𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
87notbid 319 . . . . 5 (𝑠 = 𝑆 → (¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
98raleqbi1dv 3409 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
109elrab 3684 . . 3 (𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
114, 10syl6bb 288 . 2 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
12 elfvex 6700 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝑋 ∈ V)
13 elpw2g 5244 . . . 4 (𝑋 ∈ V → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1412, 13syl 17 . . 3 (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1514anbi1d 629 . 2 (𝐴 ∈ (Moore‘𝑋) → ((𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
1611, 15bitrd 280 1 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  cdif 3937  wss 3940  𝒫 cpw 4542  {csn 4564  cfv 6352  Moorecmre 16843  mrClscmrc 16844  mrIndcmri 16845
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fv 6360  df-mre 16847  df-mri 16849
This theorem is referenced by:  ismri2  16893  mriss  16896  lbsacsbs  19848
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