Theorem mrefg2 43671

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 17771	. . . . . . . 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 17768	. . . . . . . . . 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 3941	. . . . . . . 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 3455	. . . . . . . 8 ⊢ 𝑔 ∈ V
9	8	elpw 4561	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ⊆ 𝑋)
10	8	elpw 4561	. . . . . . 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 3915	. . . . 5 ⊢ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin))
14		elin 3915	. . . . 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 4571	. . . . . . 7 ⊢ (𝑆 = (𝐹‘𝑔) → 𝒫 𝑆 = 𝒫 (𝐹‘𝑔))
17	16	ineq1d 4165	. . . . . 6 ⊢ (𝑆 = (𝐹‘𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹‘𝑔) ∩ Fin))
18	17	eleq2d 2847	. . . . 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 3183	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 2145  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ‘cfv 6531  Fincfn 8957  Moorecmre 17732  mrClscmrc 17733

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-mre 17736  df-mrc 17737

This theorem is used by:  mrefg3  43672  isnacs3  43674