Theorem mrefg3 43296

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 43295	. . 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 3954	. . . 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 4191	. . . . . . . . 9 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
7	6	sseli 3935	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
8	7	elpwid 4567	. . . . . . 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 17664	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑆 ∧ 𝑆 ∈ 𝐶) → (𝐹‘𝑔) ⊆ 𝑆)
12	5, 9, 10, 11	syl3anc 1394	. . . . 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 1563   ∈ wcel 2145  ∃wrex 3089   ∩ cin 3906   ⊆ wss 3907  𝒫 cpw 4558  ‘cfv 6525  Fincfn 8931  Moorecmre 17622  mrClscmrc 17623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-br 5105  df-opab 5167  df-mpt 5186  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 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-mre 17626  df-mrc 17627

This theorem is referenced by: (None)