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Theorem mrefg2 36750
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 16198 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → 𝑔 ⊆ (𝐹𝑔))
3 simpr 477 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔 ⊆ (𝐹𝑔))
41mrcssv 16195 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑔) ⊆ 𝑋)
54adantr 481 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → (𝐹𝑔) ⊆ 𝑋)
63, 5sstrd 3593 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔𝑋)
72, 6impbida 876 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝑔𝑋𝑔 ⊆ (𝐹𝑔)))
8 vex 3189 . . . . . . . 8 𝑔 ∈ V
98elpw 4136 . . . . . . 7 (𝑔 ∈ 𝒫 𝑋𝑔𝑋)
108elpw 4136 . . . . . . 7 (𝑔 ∈ 𝒫 (𝐹𝑔) ↔ 𝑔 ⊆ (𝐹𝑔))
117, 9, 103bitr4g 303 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ 𝒫 𝑋𝑔 ∈ 𝒫 (𝐹𝑔)))
1211anbi1d 740 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹𝑔) ∧ 𝑔 ∈ Fin)))
13 elin 3774 . . . . 5 (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin))
14 elin 3774 . . . . 5 (𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹𝑔) ∧ 𝑔 ∈ Fin))
1512, 13, 143bitr4g 303 . . . 4 (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin)))
16 pweq 4133 . . . . . . 7 (𝑆 = (𝐹𝑔) → 𝒫 𝑆 = 𝒫 (𝐹𝑔))
1716ineq1d 3791 . . . . . 6 (𝑆 = (𝐹𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹𝑔) ∩ Fin))
1817eleq2d 2684 . . . . 5 (𝑆 = (𝐹𝑔) → (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin)))
1918bibi2d 332 . . . 4 (𝑆 = (𝐹𝑔) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) ↔ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin))))
2015, 19syl5ibrcom 237 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑆 = (𝐹𝑔) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
2120pm5.32rd 671 . 2 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (𝐹𝑔)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (𝐹𝑔))))
2221rexbidv2 3041 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 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130  cfv 5847  Fincfn 7899  Moorecmre 16163  mrClscmrc 16164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-mre 16167  df-mrc 16168
This theorem is referenced by:  mrefg3  36751  isnacs3  36753
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