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Theorem mrcidmd 17576
Description: Moore closure is idempotent. Deduction form of mrcidm 17569. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrcssidd.1 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
mrcssidd.2 𝑁 = (mrClsβ€˜π΄)
mrcssidd.3 (πœ‘ β†’ π‘ˆ βŠ† 𝑋)
Assertion
Ref Expression
mrcidmd (πœ‘ β†’ (π‘β€˜(π‘β€˜π‘ˆ)) = (π‘β€˜π‘ˆ))

Proof of Theorem mrcidmd
StepHypRef Expression
1 mrcssidd.1 . 2 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
2 mrcssidd.3 . 2 (πœ‘ β†’ π‘ˆ βŠ† 𝑋)
3 mrcssidd.2 . . 3 𝑁 = (mrClsβ€˜π΄)
43mrcidm 17569 . 2 ((𝐴 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (π‘β€˜(π‘β€˜π‘ˆ)) = (π‘β€˜π‘ˆ))
51, 2, 4syl2anc 583 1 (πœ‘ β†’ (π‘β€˜(π‘β€˜π‘ˆ)) = (π‘β€˜π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6536  Moorecmre 17532  mrClscmrc 17533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-mre 17536  df-mrc 17537
This theorem is referenced by:  mressmrcd  17577  mreexexlem2d  17595  acsmap2d  18517
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