Theorem isnacs 43160

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 6869	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝑋 ∈ V)
2		elfvex 6869	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
3	2	adantr 481	. 2 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)) → 𝑋 ∈ V)
4		fveq2 6834	. . . . . 6 ⊢ (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
5		pweq 4550	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
6	5	ineq1d 4155	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
7	6	rexeqdv 3299	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
8	7	ralbidv 3163	. . . . . 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 3410	. . . . 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 43159	. . . . 5 ⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
11		fvex 6847	. . . . . 6 ⊢ (ACS‘𝑋) ∈ V
12	11	rabex 5274	. . . . 5 ⊢ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ∈ V
13	9, 10, 12	fvmpt 6942	. . . 4 ⊢ (𝑋 ∈ V → (NoeACS‘𝑋) = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
14	13	eleq2d 2826	. . 3 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}))
15		fveq2 6834	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = (mrCls‘𝐶))
16		isnacs.f	. . . . . . . . 9 ⊢ 𝐹 = (mrCls‘𝐶)
17	15, 16	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = 𝐹)
18	17	fveq1d 6836	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((mrCls‘𝑐)‘𝑔) = (𝐹‘𝑔))
19	18	eqeq2d 2751	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ 𝑠 = (𝐹‘𝑔)))
20	19	rexbidv 3164	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
21	20	raleqbi1dv 3308	. . . 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 3636	. . 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 288	. 2 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔))))
24	1, 3, 23	pm5.21nii 379	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  {crab 3392  Vcvv 3432   ∩ cin 3889  𝒫 cpw 4536  ‘cfv 6492  Fincfn 8890  mrClscmrc 17543  ACScacs 17545  NoeACScnacs 43158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-nacs 43159

This theorem is referenced by:  nacsfg  43161  isnacs2  43162  isnacs3  43166  islnr3  43567