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Theorem mrefg2 37789
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 16484 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → 𝑔 ⊆ (𝐹𝑔))
3 simpr 471 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔 ⊆ (𝐹𝑔))
41mrcssv 16481 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑔) ⊆ 𝑋)
54adantr 466 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → (𝐹𝑔) ⊆ 𝑋)
63, 5sstrd 3760 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹𝑔)) → 𝑔𝑋)
72, 6impbida 794 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝑔𝑋𝑔 ⊆ (𝐹𝑔)))
8 vex 3352 . . . . . . . 8 𝑔 ∈ V
98elpw 4301 . . . . . . 7 (𝑔 ∈ 𝒫 𝑋𝑔𝑋)
108elpw 4301 . . . . . . 7 (𝑔 ∈ 𝒫 (𝐹𝑔) ↔ 𝑔 ⊆ (𝐹𝑔))
117, 9, 103bitr4g 303 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ 𝒫 𝑋𝑔 ∈ 𝒫 (𝐹𝑔)))
1211anbi1d 607 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹𝑔) ∧ 𝑔 ∈ Fin)))
13 elin 3945 . . . . 5 (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋𝑔 ∈ Fin))
14 elin 3945 . . . . 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 4298 . . . . . . 7 (𝑆 = (𝐹𝑔) → 𝒫 𝑆 = 𝒫 (𝐹𝑔))
1716ineq1d 3962 . . . . . 6 (𝑆 = (𝐹𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹𝑔) ∩ Fin))
1817eleq2d 2835 . . . . 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 237 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑆 = (𝐹𝑔) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
2120pm5.32rd 559 . 2 (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (𝐹𝑔)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (𝐹𝑔))))
2221rexbidv2 3195 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 382   = wceq 1630  wcel 2144  wrex 3061  cin 3720  wss 3721  𝒫 cpw 4295  cfv 6031  Fincfn 8108  Moorecmre 16449  mrClscmrc 16450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-mre 16453  df-mrc 16454
This theorem is referenced by:  mrefg3  37790  isnacs3  37792
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