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Theorem predidm 5845
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 5823 . 2 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
2 df-pred 5823 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 inidm 3971 . . . . . 6 ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})) = (𝑅 “ {𝑋})
43ineq2i 3962 . . . . 5 (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋}))) = (𝐴 ∩ (𝑅 “ {𝑋}))
52, 4eqtr4i 2796 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
6 inass 3972 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
75, 6eqtr4i 2796 . . 3 Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
82ineq1i 3961 . . 3 (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
97, 8eqtr4i 2796 . 2 Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
101, 9eqtr4i 2796 1 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cin 3722  {csn 4316  ccnv 5248  cima 5252  Predcpred 5822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-pred 5823
This theorem is referenced by: (None)
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