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Theorem mrefg2 36288
Description: Slight variation on finite genration 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 16100 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → 𝑔 ⊆ (𝐹𝑔))
3 simpr 476 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔 ⊆ (𝐹𝑔))
41mrcssv 16097 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑔) ⊆ 𝑋)
54adantr 480 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → (𝐹𝑔) ⊆ 𝑋)
63, 5sstrd 3578 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔𝑋)
72, 6impbida 873 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝑔𝑋𝑔 ⊆ (𝐹𝑔)))
8 vex 3176 . . . . . . . 8 𝑔 ∈ V
98elpw 4114 . . . . . . 7 (𝑔 ∈ 𝒫 𝑋𝑔𝑋)
108elpw 4114 . . . . . . 7 (𝑔 ∈ 𝒫 (𝐹𝑔) ↔ 𝑔 ⊆ (𝐹𝑔))
117, 9, 103bitr4g 302 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ 𝒫 𝑋𝑔 ∈ 𝒫 (𝐹𝑔)))
1211anbi1d 737 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹𝑔) ∧ 𝑔 ∈ Fin)))
13 elin 3758 . . . . 5 (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin))
14 elin 3758 . . . . 5 (𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹𝑔) ∧ 𝑔 ∈ Fin))
1512, 13, 143bitr4g 302 . . . 4 (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin)))
16 pweq 4111 . . . . . . 7 (𝑆 = (𝐹𝑔) → 𝒫 𝑆 = 𝒫 (𝐹𝑔))
1716ineq1d 3775 . . . . . 6 (𝑆 = (𝐹𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹𝑔) ∩ Fin))
1817eleq2d 2673 . . . . 5 (𝑆 = (𝐹𝑔) → (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin)))
1918bibi2d 331 . . . 4 (𝑆 = (𝐹𝑔) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) ↔ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹𝑔) ∩ Fin))))
2015, 19syl5ibrcom 236 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑆 = (𝐹𝑔) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
2120pm5.32rd 670 . 2 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (𝐹𝑔)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (𝐹𝑔))))
2221rexbidv2 3030 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 195  wa 383   = wceq 1475  wcel 1977  wrex 2897  cin 3539  wss 3540  𝒫 cpw 4108  cfv 5804  Fincfn 7841  Moorecmre 16065  mrClscmrc 16066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070
This theorem is referenced by:  mrefg3  36289  isnacs3  36291
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