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Theorem predun 6267
Description: Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
Assertion
Ref Expression
predun Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predun
StepHypRef Expression
1 indir 4222 . 2 ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∪ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 6238 . 2 Pred(𝑅, (𝐴𝐵), 𝑋) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
3 df-pred 6238 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
4 df-pred 6238 . . 3 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
53, 4uneq12i 4108 . 2 (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∪ (𝐵 ∩ (𝑅 “ {𝑋})))
61, 2, 53eqtr4i 2774 1 Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cun 3896  cin 3897  {csn 4573  ccnv 5619  cima 5623  Predcpred 6237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-un 3903  df-in 3905  df-pred 6238
This theorem is referenced by: (None)
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