Theorem mrcsscl 17575

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 17544	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ∈ 𝐶) → 𝑉 ⊆ 𝑋)
2	1	3adant2 1132	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → 𝑉 ⊆ 𝑋)
3		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
4	3	mrcss 17571	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
5	2, 4	syld3an3 1412	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
6	3	mrcid 17568	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑉) = 𝑉)
7	6	3adant2 1132	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑉) = 𝑉)
8	5, 7	sseqtrd 3959	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ‘cfv 6490  Moorecmre 17533  mrClscmrc 17534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-mre 17537  df-mrc 17538

This theorem is referenced by:  submrc  17583  isacs2  17608  isacs3lem  18497  mrelatlub  18517  mndind  18785  gsumwspan  18803  symggen  19434  cntzspan  19808  dprdspan  19993  subgdmdprd  20000  subgdprd  20001  dprdsn  20002  dprd2dlem1  20007  dprd2da  20008  dmdprdsplit2lem  20011  ablfac1b  20036  pgpfac1lem1  20040  pgpfac1lem5  20045  mrccss  21682  evlseu  22070  ismrcd2  43142  mrefg3  43151  isnacs3  43153