Theorem nacsfg 39293

Description: In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Hypothesis

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

Assertion

Ref	Expression
nacsfg	⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔))

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

Proof of Theorem nacsfg

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnacs.f	. . . . 5 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	isnacs 39292	. . . 4 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
3	2	simprbi 499	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔))
4		eqeq1 2823	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 = (𝐹‘𝑔) ↔ 𝑆 = (𝐹‘𝑔)))
5	4	rexbidv 3295	. . . 4 ⊢ (𝑠 = 𝑆 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔)))
6	5	rspcva 3619	. . 3 ⊢ ((𝑆 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔))
7	3, 6	sylan2 594	. 2 ⊢ ((𝑆 ∈ 𝐶 ∧ 𝐶 ∈ (NoeACS‘𝑋)) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔))
8	7	ancoms 461	1 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  ∃wrex 3137   ∩ cin 3933  𝒫 cpw 4537  ‘cfv 6348  Fincfn 8501  mrClscmrc 16846  ACScacs 16848  NoeACScnacs 39290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-nacs 39291

This theorem is referenced by:  isnacs3  39298