Theorem mrefg2 43922

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 17823	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑋) → 𝑔 ⊆ (𝐹‘𝑔))
3		simpr 490	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → 𝑔 ⊆ (𝐹‘𝑔))
4	1	mrcssv 17820	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑔) ⊆ 𝑋)
5	4	adantr 486	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → (𝐹‘𝑔) ⊆ 𝑋)
6	3, 5	sstrd 3923	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → 𝑔 ⊆ 𝑋)
7	2, 6	impbida 813	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑔 ⊆ 𝑋 ↔ 𝑔 ⊆ (𝐹‘𝑔)))
8		vex 3458	. . . . . . . 8 ⊢ 𝑔 ∈ V
9	8	elpw 4564	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ⊆ 𝑋)
10	8	elpw 4564	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 (𝐹‘𝑔) ↔ 𝑔 ⊆ (𝐹‘𝑔))
11	7, 9, 10	3bitr4g 317	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ∈ 𝒫 (𝐹‘𝑔)))
12	11	anbi1d 643	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹‘𝑔) ∧ 𝑔 ∈ Fin)))
13		elin 4096	. . . . 5 ⊢ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin))
14		elin 4096	. . . . 5 ⊢ (𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹‘𝑔) ∧ 𝑔 ∈ Fin))
15	12, 13, 14	3bitr4g 317	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin)))
16		pweq 4574	. . . . . . 7 ⊢ (𝑆 = (𝐹‘𝑔) → 𝒫 𝑆 = 𝒫 (𝐹‘𝑔))
17	16	ineq1d 4119	. . . . . 6 ⊢ (𝑆 = (𝐹‘𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹‘𝑔) ∩ Fin))
18	17	eleq2d 2850	. . . . 5 ⊢ (𝑆 = (𝐹‘𝑔) → (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin)))
19	18	bibi2d 345	. . . 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 250	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 = (𝐹‘𝑔) → (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
21	20	pm5.32rd 589	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (𝐹‘𝑔)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (𝐹‘𝑔))))
22	21	rexbidv2 3186	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ‘cfv 6561  Fincfn 9004  Moorecmre 17784  mrClscmrc 17785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17788  df-mrc 17789

This theorem is used by:  mrefg3  43923  isnacs3  43925