Theorem mrefg3 43672

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 43671	. . 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 3946	. . . 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 4182	. . . . . . . . 9 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ 𝒫 𝑆
7	6	sseli 3927	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) → 𝑔 ∈ 𝒫 𝑆)
8	7	elpwid 4566	. . . . . . 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 17774	. . . . . 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 3185	. 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 1570   ∈ wcel 2145  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ‘cfv 6531  Fincfn 8957  Moorecmre 17732  mrClscmrc 17733

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  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 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-mre 17736  df-mrc 17737

This theorem is used by: (None)