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Theorem mrefg3 39189
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 39188 . . 3 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))
32adantr 481 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))
4 simpll 763 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
5 inss1 4209 . . . . . . . . 9 (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
65sseli 3967 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
76elpwid 4556 . . . . . . 7 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔𝑆)
87adantl 482 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑔𝑆)
9 simplr 765 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑆𝐶)
101mrcsscl 16886 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑆𝑆𝐶) → (𝐹𝑔) ⊆ 𝑆)
114, 8, 9, 10syl3anc 1365 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝐹𝑔) ⊆ 𝑆)
1211biantrud 532 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 ⊆ (𝐹𝑔) ↔ (𝑆 ⊆ (𝐹𝑔) ∧ (𝐹𝑔) ⊆ 𝑆)))
13 eqss 3986 . . . 4 (𝑆 = (𝐹𝑔) ↔ (𝑆 ⊆ (𝐹𝑔) ∧ (𝐹𝑔) ⊆ 𝑆))
1412, 13syl6rbbr 291 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 = (𝐹𝑔) ↔ 𝑆 ⊆ (𝐹𝑔)))
1514rexbidva 3301 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))
163, 15bitrd 280 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 207  wa 396   = wceq 1530  wcel 2107  wrex 3144  cin 3939  wss 3940  𝒫 cpw 4542  cfv 6354  Fincfn 8503  Moorecmre 16848  mrClscmrc 16849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-mre 16852  df-mrc 16853
This theorem is referenced by: (None)
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