Theorem mrcidb2 17327

Description: A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Hypothesis

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

Assertion

Ref	Expression
mrcidb2	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) ⊆ 𝑈))

Proof of Theorem mrcidb2

Step	Hyp	Ref	Expression
1		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	mrcidb 17324	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))
3	2	adantr 481	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))
4		eqss 3936	. . 3 ⊢ ((𝐹‘𝑈) = 𝑈 ↔ ((𝐹‘𝑈) ⊆ 𝑈 ∧ 𝑈 ⊆ (𝐹‘𝑈)))
5	1	mrcssid 17326	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝐹‘𝑈))
6	5	biantrud 532	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → ((𝐹‘𝑈) ⊆ 𝑈 ↔ ((𝐹‘𝑈) ⊆ 𝑈 ∧ 𝑈 ⊆ (𝐹‘𝑈))))
7	4, 6	bitr4id 290	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → ((𝐹‘𝑈) = 𝑈 ↔ (𝐹‘𝑈) ⊆ 𝑈))
8	3, 7	bitrd 278	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ‘cfv 6433  Moorecmre 17291  mrClscmrc 17292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-mre 17295  df-mrc 17296

This theorem is referenced by:  isacs5  18266