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Theorem mrefg2 39297
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
mrefg2 (𝐶 ∈ (Moore‘𝑋) → (∃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 mrefg2
StepHypRef Expression
1 isnacs.f . . . . . . . . 9 𝐹 = (mrCls‘𝐶)
21mrcssid 16882 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → 𝑔 ⊆ (𝐹𝑔))
3 simpr 487 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔 ⊆ (𝐹𝑔))
41mrcssv 16879 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑔) ⊆ 𝑋)
54adantr 483 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → (𝐹𝑔) ⊆ 𝑋)
63, 5sstrd 3977 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔𝑋)
72, 6impbida 799 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝑔𝑋𝑔 ⊆ (𝐹𝑔)))
8 vex 3498 . . . . . . . 8 𝑔 ∈ V
98elpw 4546 . . . . . . 7 (𝑔 ∈ 𝒫 𝑋𝑔𝑋)
108elpw 4546 . . . . . . 7 (𝑔 ∈ 𝒫 (𝐹𝑔) ↔ 𝑔 ⊆ (𝐹𝑔))
117, 9, 103bitr4g 316 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ 𝒫 𝑋𝑔 ∈ 𝒫 (𝐹𝑔)))
1211anbi1d 631 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹𝑔) ∧ 𝑔 ∈ Fin)))
13 elin 4169 . . . . 5 (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin))
14 elin 4169 . . . . 5 (𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹𝑔) ∧ 𝑔 ∈ Fin))
1512, 13, 143bitr4g 316 . . . 4 (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin)))
16 pweq 4542 . . . . . . 7 (𝑆 = (𝐹𝑔) → 𝒫 𝑆 = 𝒫 (𝐹𝑔))
1716ineq1d 4188 . . . . . 6 (𝑆 = (𝐹𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹𝑔) ∩ Fin))
1817eleq2d 2898 . . . . 5 (𝑆 = (𝐹𝑔) → (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin)))
1918bibi2d 345 . . . 4 (𝑆 = (𝐹𝑔) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) ↔ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin))))
2015, 19syl5ibrcom 249 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑆 = (𝐹𝑔) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
2120pm5.32rd 580 . 2 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (𝐹𝑔)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (𝐹𝑔))))
2221rexbidv2 3295 1 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wrex 3139  cin 3935  wss 3936  𝒫 cpw 4539  cfv 6350  Fincfn 8503  Moorecmre 16847  mrClscmrc 16848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-int 4870  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-mre 16851  df-mrc 16852
This theorem is referenced by:  mrefg3  39298  isnacs3  39300
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