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Theorem preddif 6005
 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 4142 . 2 ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∖ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 5980 . 2 Pred(𝑅, (𝐴𝐵), 𝑋) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
3 df-pred 5980 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
4 df-pred 5980 . . 3 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
53, 4difeq12i 3983 . 2 (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∖ (𝐵 ∩ (𝑅 “ {𝑋})))
61, 2, 53eqtr4i 2806 1 Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1507   ∖ cdif 3822   ∩ cin 3824  {csn 4435  ◡ccnv 5399   “ cima 5403  Predcpred 5979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rab 3091  df-v 3411  df-dif 3828  df-in 3832  df-pred 5980 This theorem is referenced by:  wfrlem8  7759  frrlem13  32596
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