Theorem mrcidmd 16891

Description: Moore closure is idempotent. Deduction form of mrcidm 16884. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mrcssidd.1	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
mrcssidd.2	⊢ 𝑁 = (mrCls‘𝐴)
mrcssidd.3	⊢ (𝜑 → 𝑈 ⊆ 𝑋)

Assertion

Ref	Expression
mrcidmd	⊢ (𝜑 → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈))

Proof of Theorem mrcidmd

Step	Hyp	Ref	Expression
1		mrcssidd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
2		mrcssidd.3	. 2 ⊢ (𝜑 → 𝑈 ⊆ 𝑋)
3		mrcssidd.2	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
4	3	mrcidm 16884	. 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈))
5	1, 2, 4	syl2anc 586	1 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   ⊆ wss 3936  ‘cfv 6350  Moorecmre 16847  mrClscmrc 16848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-int 4870  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-mre 16851  df-mrc 16852

This theorem is referenced by:  mressmrcd  16892  mreexexlem2d  16910  acsmap2d  17783