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Theorem mrcidb 16483
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 16481 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)
3 simpr 471 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) = 𝑈)
41mrcssv 16482 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
54adantr 466 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ⊆ 𝑋)
63, 5eqsstr3d 3789 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝑋)
71mrccl 16479 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
86, 7syldan 579 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ∈ 𝐶)
93, 8eqeltrrd 2851 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝐶)
102, 9impbida 802 1 (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wss 3723  cfv 6030  Moorecmre 16450  mrClscmrc 16451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-mre 16454  df-mrc 16455
This theorem is referenced by:  mrcidb2  16486
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