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

Theorem dfpred3 5607
 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 3767 . 2 (𝐴 ∩ {𝑦𝑦𝑅𝑋}) = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
2 dfpred2.1 . . 3 𝑋 ∈ V
32dfpred2 5606 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
4 dfrab2 3862 . 2 {𝑦𝐴𝑦𝑅𝑋} = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
51, 3, 43eqtr4i 2642 1 Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ∩ cin 3539   class class class wbr 4583  Predcpred 5596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597 This theorem is referenced by:  dfpred3g  5608
 Copyright terms: Public domain W3C validator