Theorem mrefg3 43422

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
mrefg3	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃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 mrefg3

Step	Hyp	Ref	Expression
1		isnacs.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	mrefg2 43421	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))
3	2	adantr 485	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))
4		eqss 3953	. . . 4 ⊢ (𝑆 = (𝐹‘𝑔) ↔ (𝑆 ⊆ (𝐹‘𝑔) ∧ (𝐹‘𝑔) ⊆ 𝑆))
5		simpll 778	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
6		inss1 4190	. . . . . . . . 9 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
7	6	sseli 3934	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
8	7	elpwid 4572	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ⊆ 𝑆)
9	8	adantl 486	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑔 ⊆ 𝑆)
10		simplr 780	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑆 ∈ 𝐶)
11	1	mrcsscl 17677	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑆 ∧ 𝑆 ∈ 𝐶) → (𝐹‘𝑔) ⊆ 𝑆)
12	5, 9, 10, 11	syl3anc 1398	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝐹‘𝑔) ⊆ 𝑆)
13	12	biantrud 540	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 ⊆ (𝐹‘𝑔) ↔ (𝑆 ⊆ (𝐹‘𝑔) ∧ (𝐹‘𝑔) ⊆ 𝑆)))
14	4, 13	bitr4id 293	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 = (𝐹‘𝑔) ↔ 𝑆 ⊆ (𝐹‘𝑔)))
15	14	rexbidva 3187	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹‘𝑔)))
16	3, 15	bitrd 282	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹‘𝑔)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4563  ‘cfv 6538  Fincfn 8944  Moorecmre 17635  mrClscmrc 17636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-mre 17639  df-mrc 17640

This theorem is referenced by: (None)