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Theorem preddif 6141
 Description: Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
preddif Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem preddif
StepHypRef Expression
1 indifdir 4210 . 2 ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∖ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 6116 . 2 Pred(𝑅, (𝐴𝐵), 𝑋) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
3 df-pred 6116 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
4 df-pred 6116 . . 3 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
53, 4difeq12i 4048 . 2 (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∖ (𝐵 ∩ (𝑅 “ {𝑋})))
61, 2, 53eqtr4i 2831 1 Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∖ cdif 3878   ∩ cin 3880  {csn 4525  ◡ccnv 5518   “ cima 5522  Predcpred 6115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-pred 6116 This theorem is referenced by:  wfrlem8  7947  frrlem13  33260
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