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Theorem mrcidb2 16206
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 16203 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
32adantr 481 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
41mrcssid 16205 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ⊆ (𝐹𝑈))
54biantrud 528 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → ((𝐹𝑈) ⊆ 𝑈 ↔ ((𝐹𝑈) ⊆ 𝑈𝑈 ⊆ (𝐹𝑈))))
6 eqss 3602 . . 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 384   = wceq 1480  wcel 1987  wss 3559  cfv 5852  Moorecmre 16170  mrClscmrc 16171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-mre 16174  df-mrc 16175
This theorem is referenced by:  isacs5  17100
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