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Theorem dfpred3g 6306
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 6298 . . 3 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2 breq2 5145 . . . 4 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
32rabbidv 3434 . . 3 (𝑥 = 𝑋 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦𝑅𝑋})
41, 3eqeq12d 2742 . 2 (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}))
5 vex 3472 . . 3 𝑥 ∈ V
65dfpred3 6305 . 2 Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥}
74, 6vtoclg 3537 1 (𝑋𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3426   class class class wbr 5141  Predcpred 6293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294
This theorem is referenced by:  lrrecpred  27816  fnrelpredd  34621  wsuclem  35330
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