MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  predin Structured version   Visualization version   GIF version

Theorem predin 6152
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 4187 . 2 ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 6129 . 2 Pred(𝑅, (𝐴𝐵), 𝑋) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
3 df-pred 6129 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
4 df-pred 6129 . . 3 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
53, 4ineq12i 4170 . 2 (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ (𝐵 ∩ (𝑅 “ {𝑋})))
61, 2, 53eqtr4i 2857 1 Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cin 3917  {csn 4548  ccnv 5535  cima 5539  Predcpred 6128
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3141  df-v 3481  df-in 3925  df-pred 6129
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator