Theorem mrefg2 40529

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 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))

Distinct variable groups:   𝐶,𝑔   𝑔,𝐹   𝑆,𝑔   𝑔,𝑋

Proof of Theorem mrefg2

Step	Hyp	Ref	Expression
1		isnacs.f	. . . . . . . . 9 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	mrcssid 17326	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑋) → 𝑔 ⊆ (𝐹‘𝑔))
3		simpr 485	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → 𝑔 ⊆ (𝐹‘𝑔))
4	1	mrcssv 17323	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑔) ⊆ 𝑋)
5	4	adantr 481	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → (𝐹‘𝑔) ⊆ 𝑋)
6	3, 5	sstrd 3931	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → 𝑔 ⊆ 𝑋)
7	2, 6	impbida 798	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑔 ⊆ 𝑋 ↔ 𝑔 ⊆ (𝐹‘𝑔)))
8		vex 3436	. . . . . . . 8 ⊢ 𝑔 ∈ V
9	8	elpw 4537	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ⊆ 𝑋)
10	8	elpw 4537	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 (𝐹‘𝑔) ↔ 𝑔 ⊆ (𝐹‘𝑔))
11	7, 9, 10	3bitr4g 314	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ∈ 𝒫 (𝐹‘𝑔)))
12	11	anbi1d 630	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹‘𝑔) ∧ 𝑔 ∈ Fin)))
13		elin 3903	. . . . 5 ⊢ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin))
14		elin 3903	. . . . 5 ⊢ (𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹‘𝑔) ∧ 𝑔 ∈ Fin))
15	12, 13, 14	3bitr4g 314	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin)))
16		pweq 4549	. . . . . . 7 ⊢ (𝑆 = (𝐹‘𝑔) → 𝒫 𝑆 = 𝒫 (𝐹‘𝑔))
17	16	ineq1d 4145	. . . . . 6 ⊢ (𝑆 = (𝐹‘𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹‘𝑔) ∩ Fin))
18	17	eleq2d 2824	. . . . 5 ⊢ (𝑆 = (𝐹‘𝑔) → (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin)))
19	18	bibi2d 343	. . . 4 ⊢ (𝑆 = (𝐹‘𝑔) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) ↔ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin))))
20	15, 19	syl5ibrcom 246	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 = (𝐹‘𝑔) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
21	20	pm5.32rd 578	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (𝐹‘𝑔)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (𝐹‘𝑔))))
22	21	rexbidv2 3224	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ‘cfv 6433  Fincfn 8733  Moorecmre 17291  mrClscmrc 17292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-mre 17295  df-mrc 17296

This theorem is referenced by:  mrefg3  40530  isnacs3  40532