Theorem mrefg3 43813

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 43812	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))
3	2	adantr 486	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))
4		eqss 3923	. . . 4 ⊢ (𝑆 = (𝐹‘𝑔) ↔ (𝑆 ⊆ (𝐹‘𝑔) ∧ (𝐹‘𝑔) ⊆ 𝑆))
5		simpll 779	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
6		inss1 4133	. . . . . . . . 9 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
7	6	sseli 3904	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
8	7	elpwid 4565	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ⊆ 𝑆)
9	8	adantl 487	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑔 ⊆ 𝑆)
10		simplr 781	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑆 ∈ 𝐶)
11	1	mrcsscl 17800	. . . . . 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 541	. . . 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 3184	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  ‘cfv 6539  Fincfn 8977  Moorecmre 17758  mrClscmrc 17759

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-mre 17762  df-mrc 17763

This theorem is used by: (None)