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Theorem dfpred3g 5608
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 5601 . . 3 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2 breq2 4587 . . . 4 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
32rabbidv 3164 . . 3 (𝑥 = 𝑋 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦𝑅𝑋})
41, 3eqeq12d 2625 . 2 (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}))
5 vex 3176 . . 3 𝑥 ∈ V
65dfpred3 5607 . 2 Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥}
74, 6vtoclg 3239 1 (𝑋𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  {crab 2900   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:  wsuclem  31017  wsuclemOLD  31018
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