Theorem mrefg2 42702

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 17585	. . . . . . . 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 17582	. . . . . . . . . 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 3960	. . . . . . . 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 3454	. . . . . . . 8 ⊢ 𝑔 ∈ V
9	8	elpw 4570	. . . . . . 7 ⊢ (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ⊆ 𝑋)
10	8	elpw 4570	. . . . . . 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 3933	. . . . 5 ⊢ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin))
14		elin 3933	. . . . 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 4580	. . . . . . 7 ⊢ (𝑆 = (𝐹‘𝑔) → 𝒫 𝑆 = 𝒫 (𝐹‘𝑔))
17	16	ineq1d 4185	. . . . . 6 ⊢ (𝑆 = (𝐹‘𝑔) → (𝒫 𝑆 ∩ Fin) = (𝒫 (𝐹‘𝑔) ∩ Fin))
18	17	eleq2d 2815	. . . . 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 3154	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 1540   ∈ wcel 2109  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  ‘cfv 6514  Fincfn 8921  Moorecmre 17550  mrClscmrc 17551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-mre 17554  df-mrc 17555

This theorem is referenced by:  mrefg3  42703  isnacs3  42705