Theorem isnacs 40170

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 6728	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝑋 ∈ V)
2		elfvex 6728	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
3	2	adantr 484	. 2 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)) → 𝑋 ∈ V)
4		fveq2 6695	. . . . . 6 ⊢ (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
5		pweq 4515	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
6	5	ineq1d 4112	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
7	6	rexeqdv 3316	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
8	7	ralbidv 3108	. . . . . 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 3386	. . . . 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 40169	. . . . 5 ⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
11		fvex 6708	. . . . . 6 ⊢ (ACS‘𝑋) ∈ V
12	11	rabex 5210	. . . . 5 ⊢ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ∈ V
13	9, 10, 12	fvmpt 6796	. . . 4 ⊢ (𝑋 ∈ V → (NoeACS‘𝑋) = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
14	13	eleq2d 2816	. . 3 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}))
15		fveq2 6695	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = (mrCls‘𝐶))
16		isnacs.f	. . . . . . . . 9 ⊢ 𝐹 = (mrCls‘𝐶)
17	15, 16	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = 𝐹)
18	17	fveq1d 6697	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((mrCls‘𝑐)‘𝑔) = (𝐹‘𝑔))
19	18	eqeq2d 2747	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ 𝑠 = (𝐹‘𝑔)))
20	19	rexbidv 3206	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
21	20	raleqbi1dv 3307	. . . 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 3591	. . 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 290	. 2 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔))))
24	1, 3, 23	pm5.21nii 383	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052  {crab 3055  Vcvv 3398   ∩ cin 3852  𝒫 cpw 4499  ‘cfv 6358  Fincfn 8604  mrClscmrc 17040  ACScacs 17042  NoeACScnacs 40168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6316  df-fun 6360  df-fv 6366  df-nacs 40169

This theorem is referenced by:  nacsfg  40171  isnacs2  40172  isnacs3  40176  islnr3  40584