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Theorem predidm 6151
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 6129 . 2 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
2 df-pred 6129 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 inidm 4109 . . . . . 6 ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})) = (𝑅 “ {𝑋})
43ineq2i 4100 . . . . 5 (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋}))) = (𝐴 ∩ (𝑅 “ {𝑋}))
52, 4eqtr4i 2764 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
6 inass 4110 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
75, 6eqtr4i 2764 . . 3 Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
82ineq1i 4099 . . 3 (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
97, 8eqtr4i 2764 . 2 Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
101, 9eqtr4i 2764 1 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cin 3842  {csn 4516  ccnv 5524  cima 5528  Predcpred 6128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-in 3850  df-pred 6129
This theorem is referenced by: (None)
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