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Theorem dfpred3g 6335
Description: An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
dfpred3g (𝑋𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋})
Distinct variable groups:   𝑦,𝑅   𝑦,𝐴   𝑦,𝑋
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem dfpred3g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 predeq3 6327 . . 3 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2 breq2 5152 . . . 4 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
32rabbidv 3441 . . 3 (𝑥 = 𝑋 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦𝑅𝑋})
41, 3eqeq12d 2751 . 2 (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}))
5 vex 3482 . . 3 𝑥 ∈ V
65dfpred3 6334 . 2 Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥}
74, 6vtoclg 3554 1 (𝑋𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433   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:  lrrecpred  27992  fnrelpredd  35082  wsuclem  35807
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