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Theorem mrefg3 37588
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 37587 . . 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 simpll 805 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
5 inss1 3866 . . . . . . . . 9 (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
65sseli 3632 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
76elpwid 4203 . . . . . . 7 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔𝑆)
87adantl 481 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑔𝑆)
9 simplr 807 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑆𝐶)
101mrcsscl 16327 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑆𝑆𝐶) → (𝐹𝑔) ⊆ 𝑆)
114, 8, 9, 10syl3anc 1366 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝐹𝑔) ⊆ 𝑆)
1211biantrud 527 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 ⊆ (𝐹𝑔) ↔ (𝑆 ⊆ (𝐹𝑔) ∧ (𝐹𝑔) ⊆ 𝑆)))
13 eqss 3651 . . . 4 (𝑆 = (𝐹𝑔) ↔ (𝑆 ⊆ (𝐹𝑔) ∧ (𝐹𝑔) ⊆ 𝑆))
1412, 13syl6rbbr 279 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 = (𝐹𝑔) ↔ 𝑆 ⊆ (𝐹𝑔)))
1514rexbidva 3078 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))
163, 15bitrd 268 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 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191  cfv 5926  Fincfn 7997  Moorecmre 16289  mrClscmrc 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-mre 16293  df-mrc 16294
This theorem is referenced by: (None)
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