Theorem ismri 17568

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) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 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.

Step	Hyp	Ref	Expression
1		ismri.1	. . . . 5 ⊢ 𝑁 = (mrCls‘𝐴)
2		ismri.2	. . . . 5 ⊢ 𝐼 = (mrInd‘𝐴)
3	1, 2	mrisval 17567	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
4	3	eleq2d 2814	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}))
5		difeq1 4078	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
6	5	fveq2d 6844	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑁‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥})))
7	6	eleq2d 2814	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
8	7	notbid 318	. . . . 5 ⊢ (𝑠 = 𝑆 → (¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
9	8	raleqbi1dv 3308	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
10	9	elrab 3656	. . 3 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
11	4, 10	bitrdi 287	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
12		elfvex 6878	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝑋 ∈ V)
13		elpw2g 5283	. . . 4 ⊢ (𝑋 ∈ V → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
14	12, 13	syl 17	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
15	14	anbi1d 631	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → ((𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
16	11, 15	bitrd 279	1 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402  Vcvv 3444   ∖ cdif 3908   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ‘cfv 6499  Moorecmre 17519  mrClscmrc 17520  mrIndcmri 17521

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fv 6507  df-mre 17523  df-mri 17525

This theorem is referenced by:  ismri2  17569  mriss  17572  lbsacsbs  21042