Theorem mrcsscl 17310

Description: The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
mrcsscl	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝑉)

Proof of Theorem mrcsscl

Step	Hyp	Ref	Expression
1		mress 17283	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ∈ 𝐶) → 𝑉 ⊆ 𝑋)
2	1	3adant2 1129	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → 𝑉 ⊆ 𝑋)
3		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
4	3	mrcss 17306	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
5	2, 4	syld3an3 1407	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
6	3	mrcid 17303	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑉) = 𝑉)
7	6	3adant2 1129	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑉) = 𝑉)
8	5, 7	sseqtrd 3965	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109   ⊆ wss 3891  ‘cfv 6430  Moorecmre 17272  mrClscmrc 17273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-int 4885  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-mre 17276  df-mrc 17277

This theorem is referenced by:  submrc  17318  isacs2  17343  isacs3lem  18241  mrelatlub  18261  mndind  18447  gsumwspan  18466  symggen  19059  cntzspan  19426  dprdspan  19611  subgdmdprd  19618  subgdprd  19619  dprdsn  19620  dprd2dlem1  19625  dprd2da  19626  dmdprdsplit2lem  19629  ablfac1b  19654  pgpfac1lem1  19658  pgpfac1lem5  19663  mrccss  20880  evlseu  21274  ismrcd2  40501  mrefg3  40510  isnacs3  40512