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

Theorem predpredss 6301
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 4228 . 2 (𝐴𝐵 → (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 6294 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 df-pred 6294 . 2 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
41, 2, 33sstr4g 4022 1 (𝐴𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3942  wss 3943  {csn 4623  ccnv 5668  cima 5672  Predcpred 6293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-pred 6294
This theorem is referenced by:  preddowncl  6327  wfrlem8OLD  8317
  Copyright terms: Public domain W3C validator