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Theorem predin 5672
Description: Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
Assertion
Ref Expression
predin Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predin
StepHypRef Expression
1 inindir 3815 . 2 ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 5649 . 2 Pred(𝑅, (𝐴𝐵), 𝑋) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
3 df-pred 5649 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
4 df-pred 5649 . . 3 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
53, 4ineq12i 3796 . 2 (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝐵 ∩ (𝑅 “ {𝑋})))
61, 2, 53eqtr4i 2653 1 Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cin 3559  {csn 4155  ccnv 5083  cima 5087  Predcpred 5648
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 3192  df-in 3567  df-pred 5649
This theorem is referenced by: (None)
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