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Theorem setlikespec 5919
Description: If 𝑅 is set-like in 𝐴, then all predecessors 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 vex 3388 . . . . . 6 𝑥 ∈ V
21elpred 5911 . . . . 5 (𝑋𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥𝐴𝑥𝑅𝑋)))
32abbi2dv 2919 . . . 4 (𝑋𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)})
4 df-rab 3098 . . . 4 {𝑥𝐴𝑥𝑅𝑋} = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)}
53, 4syl6reqr 2852 . . 3 (𝑋𝐴 → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
65adantr 473 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
7 seex 5275 . . 3 ((𝑅 Se 𝐴𝑋𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
87ancoms 451 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
96, 8eqeltrrd 2879 1 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {cab 2785  {crab 3093  Vcvv 3385   class class class wbr 4843   Se wse 5269  Predcpred 5897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-se 5272  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898
This theorem is referenced by:  wfrlem15  7668  trpredtr  32242  trpredmintr  32243  trpredelss  32244  dftrpred3g  32245  trpredpo  32247  trpredrec  32250  frmin  32255
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