Theorem isnacs 41490

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 6930	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝑋 ∈ V)
2		elfvex 6930	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
3	2	adantr 482	. 2 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)) → 𝑋 ∈ V)
4		fveq2 6892	. . . . . 6 ⊢ (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
5		pweq 4617	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
6	5	ineq1d 4212	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
7	6	rexeqdv 3327	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
8	7	ralbidv 3178	. . . . . 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 3450	. . . . 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 41489	. . . . 5 ⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
11		fvex 6905	. . . . . 6 ⊢ (ACS‘𝑋) ∈ V
12	11	rabex 5333	. . . . 5 ⊢ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ∈ V
13	9, 10, 12	fvmpt 6999	. . . 4 ⊢ (𝑋 ∈ V → (NoeACS‘𝑋) = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
14	13	eleq2d 2820	. . 3 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}))
15		fveq2 6892	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = (mrCls‘𝐶))
16		isnacs.f	. . . . . . . . 9 ⊢ 𝐹 = (mrCls‘𝐶)
17	15, 16	eqtr4di 2791	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (mrCls‘𝑐) = 𝐹)
18	17	fveq1d 6894	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((mrCls‘𝑐)‘𝑔) = (𝐹‘𝑔))
19	18	eqeq2d 2744	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ 𝑠 = (𝐹‘𝑔)))
20	19	rexbidv 3179	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
21	20	raleqbi1dv 3334	. . . 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 3684	. . 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 287	. 2 ⊢ (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔))))
24	1, 3, 23	pm5.21nii 380	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   ∩ cin 3948  𝒫 cpw 4603  ‘cfv 6544  Fincfn 8939  mrClscmrc 17527  ACScacs 17529  NoeACScnacs 41488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-nacs 41489

This theorem is referenced by:  nacsfg  41491  isnacs2  41492  isnacs3  41496  islnr3  41905