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Theorem mrcidb2 16480
 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) ∧ 𝑈𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) ⊆ 𝑈))

Proof of Theorem mrcidb2
StepHypRef Expression
1 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
21mrcidb 16477 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
32adantr 472 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
41mrcssid 16479 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ⊆ (𝐹𝑈))
54biantrud 529 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → ((𝐹𝑈) ⊆ 𝑈 ↔ ((𝐹𝑈) ⊆ 𝑈𝑈 ⊆ (𝐹𝑈))))
6 eqss 3759 . . 3 ((𝐹𝑈) = 𝑈 ↔ ((𝐹𝑈) ⊆ 𝑈𝑈 ⊆ (𝐹𝑈)))
75, 6syl6rbbr 279 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → ((𝐹𝑈) = 𝑈 ↔ (𝐹𝑈) ⊆ 𝑈))
83, 7bitrd 268 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) ⊆ 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ⊆ wss 3715  ‘cfv 6049  Moorecmre 16444  mrClscmrc 16445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-mre 16448  df-mrc 16449 This theorem is referenced by:  isacs5  17373
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