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Theorem predidm 6168
Description: Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)
Assertion
Ref Expression
predidm Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)

Proof of Theorem predidm
StepHypRef Expression
1 df-pred 6146 . 2 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
2 df-pred 6146 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 inidm 4199 . . . . . 6 ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})) = (𝑅 “ {𝑋})
43ineq2i 4190 . . . . 5 (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋}))) = (𝐴 ∩ (𝑅 “ {𝑋}))
52, 4eqtr4i 2852 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
6 inass 4200 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
75, 6eqtr4i 2852 . . 3 Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
82ineq1i 4189 . . 3 (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
97, 8eqtr4i 2852 . 2 Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
101, 9eqtr4i 2852 1 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  cin 3939  {csn 4564  ccnv 5553  cima 5557  Predcpred 6145
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-in 3947  df-pred 6146
This theorem is referenced by: (None)
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