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Theorem setlikespec 5670
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 3193 . . . . . 6 𝑥 ∈ V
21elpred 5662 . . . . 5 (𝑋𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥𝐴𝑥𝑅𝑋)))
32abbi2dv 2739 . . . 4 (𝑋𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)})
4 df-rab 2917 . . . 4 {𝑥𝐴𝑥𝑅𝑋} = {𝑥 ∣ (𝑥𝐴𝑥𝑅𝑋)}
53, 4syl6reqr 2674 . . 3 (𝑋𝐴 → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
65adantr 481 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
7 seex 5047 . . 3 ((𝑅 Se 𝐴𝑋𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
87ancoms 469 . 2 ((𝑋𝐴𝑅 Se 𝐴) → {𝑥𝐴𝑥𝑅𝑋} ∈ V)
96, 8eqeltrrd 2699 1 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2912  Vcvv 3190   class class class wbr 4623   Se wse 5041  Predcpred 5648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-se 5044  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649
This theorem is referenced by:  wfrlem15  7389  trpredtr  31484  trpredmintr  31485  trpredelss  31486  dftrpred3g  31487  trpredpo  31489  trpredrec  31492  frmin  31493
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