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Theorem dfpred3g 6332
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 6324 . . 3 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2 breq2 5146 . . . 4 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
32rabbidv 3443 . . 3 (𝑥 = 𝑋 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦𝑅𝑋})
41, 3eqeq12d 2752 . 2 (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}))
5 vex 3483 . . 3 𝑥 ∈ V
65dfpred3 6331 . 2 Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥}
74, 6vtoclg 3553 1 (𝑋𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {crab 3435   class class class wbr 5142  Predcpred 6319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320
This theorem is referenced by:  lrrecpred  27978  fnrelpredd  35104  wsuclem  35827
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