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Theorem setlikespec 6327
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 3434 . . . 4 {𝑥𝐴𝑥𝑅𝑋} = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)}
2 vex 3479 . . . . . 6 𝑥 ∈ V
32elpred 6318 . . . . 5 (𝑋𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥𝐴𝑥𝑅𝑋)))
43eqabdv 2868 . . . 4 (𝑋𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)})
51, 4eqtr4id 2792 . . 3 (𝑋𝐴 → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
65adantr 482 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
7 seex 5639 . . 3 ((𝑅 Se 𝐴𝑋𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
87ancoms 460 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
96, 8eqeltrrd 2835 1 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  {crab 3433  Vcvv 3475   class class class wbr 5149   Se wse 5630  Predcpred 6300
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-se 5633  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301
This theorem is referenced by:  sexp2  8132  sexp3  8139  fpr1  8288  wfrlem15OLD  8323  ttrclselem2  9721  frmin  9744  frr1  9754
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