Theorem mrcsscl 16994

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 16967	. . . 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 16990	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
5	2, 4	syld3an3 1410	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
6	3	mrcid 16987	. . 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 3917	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ⊆ wss 3843  ‘cfv 6339  Moorecmre 16956  mrClscmrc 16957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-mre 16960  df-mrc 16961

This theorem is referenced by:  submrc  17002  isacs2  17027  isacs3lem  17892  mrelatlub  17912  mndind  18108  gsumwspan  18127  symggen  18716  cntzspan  19083  dprdspan  19268  subgdmdprd  19275  subgdprd  19276  dprdsn  19277  dprd2dlem1  19282  dprd2da  19283  dmdprdsplit2lem  19286  ablfac1b  19311  pgpfac1lem1  19315  pgpfac1lem5  19320  mrccss  20510  evlseu  20897  ismrcd2  40093  mrefg3  40102  isnacs3  40104