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Theorem predidm 6229
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 6202 . 2 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
2 df-pred 6202 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 inidm 4152 . . . . . 6 ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})) = (𝑅 “ {𝑋})
43ineq2i 4143 . . . . 5 (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋}))) = (𝐴 ∩ (𝑅 “ {𝑋}))
52, 4eqtr4i 2769 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
6 inass 4153 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
75, 6eqtr4i 2769 . . 3 Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
82ineq1i 4142 . . 3 (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
97, 8eqtr4i 2769 . 2 Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
101, 9eqtr4i 2769 1 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cin 3886  {csn 4561  ccnv 5588  cima 5592  Predcpred 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-pred 6202
This theorem is referenced by: (None)
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