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Theorem mrefg3 42719
Description: Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrefg3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))
Distinct variable groups:   𝐶,𝑔   𝑔,𝐹   𝑆,𝑔   𝑔,𝑋

Proof of Theorem mrefg3
StepHypRef Expression
1 isnacs.f . . . 4 𝐹 = (mrCls‘𝐶)
21mrefg2 42718 . . 3 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))
32adantr 480 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))
4 eqss 3999 . . . 4 (𝑆 = (𝐹𝑔) ↔ (𝑆 ⊆ (𝐹𝑔) ∧ (𝐹𝑔) ⊆ 𝑆))
5 simpll 767 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
6 inss1 4237 . . . . . . . . 9 (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
76sseli 3979 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
87elpwid 4609 . . . . . . 7 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔𝑆)
98adantl 481 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑔𝑆)
10 simplr 769 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑆𝐶)
111mrcsscl 17663 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑆𝑆𝐶) → (𝐹𝑔) ⊆ 𝑆)
125, 9, 10, 11syl3anc 1373 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝐹𝑔) ⊆ 𝑆)
1312biantrud 531 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 ⊆ (𝐹𝑔) ↔ (𝑆 ⊆ (𝐹𝑔) ∧ (𝐹𝑔) ⊆ 𝑆)))
144, 13bitr4id 290 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 = (𝐹𝑔) ↔ 𝑆 ⊆ (𝐹𝑔)))
1514rexbidva 3177 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))
163, 15bitrd 279 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600  cfv 6561  Fincfn 8985  Moorecmre 17625  mrClscmrc 17626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630
This theorem is referenced by: (None)
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