Theorem ismri 17650

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 17649	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
4	3	eleq2d 2819	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}))
5		difeq1 4101	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
6	5	fveq2d 6891	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑁‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥})))
7	6	eleq2d 2819	. . . . . 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 3322	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
10	9	elrab 3676	. . 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 6925	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝑋 ∈ V)
13		elpw2g 5315	. . . 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 1539   ∈ wcel 2107  ∀wral 3050  {crab 3420  Vcvv 3464   ∖ cdif 3930   ⊆ wss 3933  𝒫 cpw 4582  {csn 4608  ‘cfv 6542  Moorecmre 17601  mrClscmrc 17602  mrIndcmri 17603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6495  df-fun 6544  df-fv 6550  df-mre 17605  df-mri 17607

This theorem is referenced by:  ismri2  17651  mriss  17654  lbsacsbs  21131