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Theorem ismri 16499
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 16498 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
43eleq2d 2836 . . 3 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}))
5 difeq1 3872 . . . . . . . 8 (𝑠 = 𝑆 → (𝑠 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
65fveq2d 6337 . . . . . . 7 (𝑠 = 𝑆 → (𝑁‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥})))
76eleq2d 2836 . . . . . 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 3295 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
109elrab 3515 . . 3 (𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
114, 10syl6bb 276 . 2 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
12 elfvex 6364 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝑋 ∈ V)
13 elpw2g 4959 . . . 4 (𝑋 ∈ V → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1412, 13syl 17 . . 3 (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1514anbi1d 615 . 2 (𝐴 ∈ (Moore‘𝑋) → ((𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
1611, 15bitrd 268 1 (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cdif 3720  wss 3723  𝒫 cpw 4298  {csn 4317  cfv 6030  Moorecmre 16450  mrClscmrc 16451  mrIndcmri 16452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fv 6038  df-mre 16454  df-mri 16456
This theorem is referenced by:  ismri2  16500  mriss  16503  lbsacsbs  19371
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