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Theorem setlikespec 5600
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 3171 . . . . . 6 𝑥 ∈ V
21elpred 5592 . . . . 5 (𝑋𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥𝐴𝑥𝑅𝑋)))
32abbi2dv 2724 . . . 4 (𝑋𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)})
4 df-rab 2900 . . . 4 {𝑥𝐴𝑥𝑅𝑋} = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)}
53, 4syl6reqr 2658 . . 3 (𝑋𝐴 → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
65adantr 479 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
7 seex 4987 . . 3 ((𝑅 Se 𝐴𝑋𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
87ancoms 467 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
96, 8eqeltrrd 2684 1 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  {cab 2591  {crab 2895  Vcvv 3168   class class class wbr 4573   Se wse 4981  Predcpred 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-se 4984  df-xp 5030  df-cnv 5032  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579
This theorem is referenced by:  wfrlem15  7289  trpredtr  30776  trpredmintr  30777  trpredelss  30778  dftrpred3g  30779  trpredpo  30781  trpredrec  30784  frmin  30785
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