Theorem mrefg3 42696

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 42695	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))
3	2	adantr 480	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))
4		eqss 3962	. . . 4 ⊢ (𝑆 = (𝐹‘𝑔) ↔ (𝑆 ⊆ (𝐹‘𝑔) ∧ (𝐹‘𝑔) ⊆ 𝑆))
5		simpll 766	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
6		inss1 4200	. . . . . . . . 9 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
7	6	sseli 3942	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
8	7	elpwid 4572	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ⊆ 𝑆)
9	8	adantl 481	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑔 ⊆ 𝑆)
10		simplr 768	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑆 ∈ 𝐶)
11	1	mrcsscl 17581	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑆 ∧ 𝑆 ∈ 𝐶) → (𝐹‘𝑔) ⊆ 𝑆)
12	5, 9, 10, 11	syl3anc 1373	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝐹‘𝑔) ⊆ 𝑆)
13	12	biantrud 531	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 ⊆ (𝐹‘𝑔) ↔ (𝑆 ⊆ (𝐹‘𝑔) ∧ (𝐹‘𝑔) ⊆ 𝑆)))
14	4, 13	bitr4id 290	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → (𝑆 = (𝐹‘𝑔) ↔ 𝑆 ⊆ (𝐹‘𝑔)))
15	14	rexbidva 3155	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹‘𝑔)))
16	3, 15	bitrd 279	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ‘cfv 6511  Fincfn 8918  Moorecmre 17543  mrClscmrc 17544

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 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-mre 17547  df-mrc 17548

This theorem is referenced by: (None)