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Theorem setlikespec 6338
Description: If 𝑅 is set-like in 𝐴, then all predecessor classes of elements of 𝐴 exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
setlikespec ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Proof of Theorem setlikespec
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-rab 3420 . . . 4 {𝑥𝐴𝑥𝑅𝑋} = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)}
2 vex 3466 . . . . . 6 𝑥 ∈ V
32elpred 6329 . . . . 5 (𝑋𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥𝐴𝑥𝑅𝑋)))
43eqabdv 2860 . . . 4 (𝑋𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)})
51, 4eqtr4id 2785 . . 3 (𝑋𝐴 → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
65adantr 479 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
7 seex 5644 . . 3 ((𝑅 Se 𝐴𝑋𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
87ancoms 457 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
96, 8eqeltrrd 2827 1 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {cab 2703  {crab 3419  Vcvv 3462   class class class wbr 5153   Se wse 5635  Predcpred 6311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-se 5638  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312
This theorem is referenced by:  sexp2  8160  sexp3  8167  fpr1  8318  wfrlem15OLD  8353  ttrclselem2  9769  frmin  9792  frr1  9802
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