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Theorem predidm 5663
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 5641 . 2 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
2 df-pred 5641 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 inidm 3802 . . . . . 6 ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})) = (𝑅 “ {𝑋})
43ineq2i 3791 . . . . 5 (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋}))) = (𝐴 ∩ (𝑅 “ {𝑋}))
52, 4eqtr4i 2646 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
6 inass 3803 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ ((𝑅 “ {𝑋}) ∩ (𝑅 “ {𝑋})))
75, 6eqtr4i 2646 . . 3 Pred(𝑅, 𝐴, 𝑋) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
82ineq1i 3790 . . 3 (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝑅 “ {𝑋}))
97, 8eqtr4i 2646 . 2 Pred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ (𝑅 “ {𝑋}))
101, 9eqtr4i 2646 1 Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cin 3555  {csn 4150  ccnv 5075  cima 5079  Predcpred 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3563  df-pred 5641
This theorem is referenced by: (None)
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