Theorem isnacs 42298

Description: Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Hypothesis

Ref	Expression
isnacs.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
isnacs	⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))

Distinct variable groups:   𝐶,𝑔,𝑠   𝑔,𝐹,𝑠   𝑔,𝑋,𝑠

Proof of Theorem isnacs

Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6938	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝑋 ∈ V)
2		elfvex 6938	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
3	2	adantr 479	. 2 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)) → 𝑋 ∈ V)
4		fveq2 6900	. . . . . 6 ⊢ (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
5		pweq 4620	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
6	5	ineq1d 4211	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
7	6	rexeqdv 3315	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
8	7	ralbidv 3167	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
9	4, 8	rabeqbidv 3436	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
10		df-nacs 42297	. . . . 5 ⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
11		fvex 6913	. . . . . 6 ⊢ (ACS‘𝑋) ∈ V
12	11	rabex 5338	. . . . 5 ⊢ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ∈ V
13	9, 10, 12	fvmpt 7008	. . . 4 ⊢ (𝑋 ∈ V → (NoeACS‘𝑋) = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
14	13	eleq2d 2811	. . 3 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}))
15		fveq2 6900	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = (mrCls‘𝐶))
16		isnacs.f	. . . . . . . . 9 ⊢ 𝐹 = (mrCls‘𝐶)
17	15, 16	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = 𝐹)
18	17	fveq1d 6902	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((mrCls‘𝑐)‘𝑔) = (𝐹‘𝑔))
19	18	eqeq2d 2736	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ 𝑠 = (𝐹‘𝑔)))
20	19	rexbidv 3168	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
21	20	raleqbi1dv 3322	. . . 4 ⊢ (𝑐 = 𝐶 → (∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
22	21	elrab 3680	. . 3 ⊢ (𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
23	14, 22	bitrdi 286	. 2 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔))))
24	1, 3, 23	pm5.21nii 377	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059  {crab 3418  Vcvv 3461   ∩ cin 3945  𝒫 cpw 4606  ‘cfv 6553  Fincfn 8973  mrClscmrc 17591  ACScacs 17593  NoeACScnacs 42296

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-iota 6505  df-fun 6555  df-fv 6561  df-nacs 42297

This theorem is referenced by:  nacsfg  42299  isnacs2  42300  isnacs3  42304  islnr3  42713