MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfpred3 Structured version   Visualization version   GIF version

Theorem dfpred3 6334
Description: An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.)
Hypothesis
Ref Expression
dfpred2.1 𝑋 ∈ V
Assertion
Ref Expression
dfpred3 Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}
Distinct variable groups:   𝑦,𝑅   𝑦,𝑋   𝑦,𝐴

Proof of Theorem dfpred3
StepHypRef Expression
1 incom 4217 . 2 (𝐴 ∩ {𝑦𝑦𝑅𝑋}) = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
2 dfpred2.1 . . 3 𝑋 ∈ V
32dfpred2 6333 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
4 dfrab2 4326 . 2 {𝑦𝐴𝑦𝑅𝑋} = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
51, 3, 43eqtr4i 2773 1 Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478  cin 3962   class class class wbr 5148  Predcpred 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323
This theorem is referenced by:  dfpred3g  6335  frpomin2  6364  fnrelpredd  35082  nummin  35084
  Copyright terms: Public domain W3C validator