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Theorem dfpred3 6311
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 4201 . 2 (𝐴 ∩ {𝑦𝑦𝑅𝑋}) = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
2 dfpred2.1 . . 3 𝑋 ∈ V
32dfpred2 6310 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
4 dfrab2 4310 . 2 {𝑦𝐴𝑦𝑅𝑋} = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
51, 3, 43eqtr4i 2770 1 Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  {cab 2709  {crab 3432  Vcvv 3474  cin 3947   class class class wbr 5148  Predcpred 6299
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300
This theorem is referenced by:  dfpred3g  6312  frpomin2  6342  fnrelpredd  34087  nummin  34089
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