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

Theorem predpredss 5650
Description: If 𝐴 is a subset of 𝐵, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predpredss (𝐴𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predpredss
StepHypRef Expression
1 ssrin 3821 . 2 (𝐴𝐵 → (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 5644 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 df-pred 5644 . 2 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
41, 2, 33sstr4g 3630 1 (𝐴𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3558  wss 3559  {csn 4153  ccnv 5078  cima 5082  Predcpred 5643
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 3191  df-in 3566  df-ss 3573  df-pred 5644
This theorem is referenced by:  preddowncl  5671  wfrlem8  7374
  Copyright terms: Public domain W3C validator