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Theorem dfpred3g 6270
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 6262 . . 3 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2 breq2 5114 . . . 4 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
32rabbidv 3418 . . 3 (𝑥 = 𝑋 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦𝑅𝑋})
41, 3eqeq12d 2753 . 2 (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}))
5 vex 3452 . . 3 𝑥 ∈ V
65dfpred3 6269 . 2 Pred(𝑅, 𝐴, 𝑥) = {𝑦𝐴𝑦𝑅𝑥}
74, 6vtoclg 3528 1 (𝑋𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3410   class class class wbr 5110  Predcpred 6257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258
This theorem is referenced by:  lrrecpred  27278  fnrelpredd  33733  wsuclem  34439
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