Theorem mrefg2 42723

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 17661	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑋) → 𝑔 ⊆ (𝐹‘𝑔))
3		simpr 484	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → 𝑔 ⊆ (𝐹‘𝑔))
4	1	mrcssv 17658	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑔) ⊆ 𝑋)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → (𝐹‘𝑔) ⊆ 𝑋)
6	3, 5	sstrd 3993	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ (𝐹‘𝑔)) → 𝑔 ⊆ 𝑋)
7	2, 6	impbida 800	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑔 ⊆ 𝑋 ↔ 𝑔 ⊆ (𝐹‘𝑔)))
8		vex 3483	. . . . . . . 8 ⊢ 𝑔 ∈ V
9	8	elpw 4603	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ⊆ 𝑋)
10	8	elpw 4603	. . . . . . 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 631	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (𝐹‘𝑔) ∧ 𝑔 ∈ Fin)))
13		elin 3966	. . . . 5 ⊢ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin))
14		elin 3966	. . . . 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 4613	. . . . . . 7 ⊢ (𝑆 = (𝐹‘𝑔) → 𝒫 𝑆 = 𝒫 (𝐹‘𝑔))
17	16	ineq1d 4218	. . . . . 6 ⊢ (𝑆 = (𝐹‘𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹‘𝑔) ∩ Fin))
18	17	eleq2d 2826	. . . . 5 ⊢ (𝑆 = (𝐹‘𝑔) → (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (𝐹‘𝑔) ∩ Fin)))
19	18	bibi2d 342	. . . 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 247	. . 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 3174	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3069   ∩ cin 3949   ⊆ wss 3950  𝒫 cpw 4599  ‘cfv 6560  Fincfn 8986  Moorecmre 17626  mrClscmrc 17627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-mre 17630  df-mrc 17631

This theorem is referenced by:  mrefg3  42724  isnacs3  42726