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Theorem mrcidb 17563
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 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrcidb (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (π‘ˆ ∈ 𝐢 ↔ (πΉβ€˜π‘ˆ) = π‘ˆ))
Proof of Theorem mrcidb
StepHypRef Expression
1 mrcfval.f . . 3 𝐹 = (mrClsβ€˜πΆ)
21mrcid 17561 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) = π‘ˆ)
3 simpr 485 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ (πΉβ€˜π‘ˆ) = π‘ˆ) β†’ (πΉβ€˜π‘ˆ) = π‘ˆ)
41mrcssv 17562 . . . . . 6 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝑋)
54adantr 481 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ (πΉβ€˜π‘ˆ) = π‘ˆ) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝑋)
63, 5eqsstrrd 4021 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ (πΉβ€˜π‘ˆ) = π‘ˆ) β†’ π‘ˆ βŠ† 𝑋)
71mrccl 17559 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) ∈ 𝐢)
86, 7syldan 591 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ (πΉβ€˜π‘ˆ) = π‘ˆ) β†’ (πΉβ€˜π‘ˆ) ∈ 𝐢)
93, 8eqeltrrd 2834 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ (πΉβ€˜π‘ˆ) = π‘ˆ) β†’ π‘ˆ ∈ 𝐢)
102, 9impbida 799 1 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (π‘ˆ ∈ 𝐢 ↔ (πΉβ€˜π‘ˆ) = π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  β€˜cfv 6543  Moorecmre 17530  mrClscmrc 17531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17534  df-mrc 17535
This theorem is referenced by:  mrcidb2  17566
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