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Theorem mrcidb 16715
 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 16713 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)
3 simpr 485 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) = 𝑈)
41mrcssv 16714 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
54adantr 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ⊆ 𝑋)
63, 5eqsstrrd 3927 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝑋)
71mrccl 16711 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
86, 7syldan 591 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ∈ 𝐶)
93, 8eqeltrrd 2884 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝐶)
102, 9impbida 797 1 (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081   ⊆ wss 3859  ‘cfv 6225  Moorecmre 16682  mrClscmrc 16683 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-mre 16686  df-mrc 16687 This theorem is referenced by:  mrcidb2  16718
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