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Theorem ismri 16114
 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 16113 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
43eleq2d 2673 . . 3 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}))
5 difeq1 3683 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
65fveq2d 6107 . . . . . . 7 (𝑠 = 𝑆 → (𝑁‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥})))
76eleq2d 2673 . . . . . 6 (𝑠 = 𝑆 → (𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
87notbid 307 . . . . 5 (𝑠 = 𝑆 → (¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
98raleqbi1dv 3123 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
109elrab 3331 . . 3 (𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
114, 10syl6bb 275 . 2 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
12 elfvex 6131 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝑋 ∈ V)
13 elpw2g 4754 . . . 4 (𝑋 ∈ V → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1412, 13syl 17 . . 3 (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1514anbi1d 737 . 2 (𝐴 ∈ (Moore‘𝑋) → ((𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
1611, 15bitrd 267 1 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  Moorecmre 16065  mrClscmrc 16066  mrIndcmri 16067 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-mre 16069  df-mri 16071 This theorem is referenced by:  ismri2  16115  mriss  16118  lbsacsbs  18977
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