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Theorem mrcidb 16256
Description: A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcidb (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))

Proof of Theorem mrcidb
StepHypRef Expression
1 mrcfval.f . . 3 𝐹 = (mrCls‘𝐶)
21mrcid 16254 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)
3 simpr 477 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) = 𝑈)
41mrcssv 16255 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
54adantr 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ⊆ 𝑋)
63, 5eqsstr3d 3632 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝑋)
71mrccl 16252 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
86, 7syldan 487 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ∈ 𝐶)
93, 8eqeltrrd 2700 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝐶)
102, 9impbida 876 1 (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wss 3567  cfv 5876  Moorecmre 16223  mrClscmrc 16224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-mre 16227  df-mrc 16228
This theorem is referenced by:  mrcidb2  16259
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