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Theorem rabfi 8070
Description: A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.)
Assertion
Ref Expression
rabfi (𝐴 ∈ Fin → {𝑥𝐴𝜑} ∈ Fin)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabfi
StepHypRef Expression
1 dfrab3 3861 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
2 infi 8069 . 2 (𝐴 ∈ Fin → (𝐴 ∩ {𝑥𝜑}) ∈ Fin)
31, 2syl5eqel 2692 1 (𝐴 ∈ Fin → {𝑥𝐴𝜑} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  {cab 2596  {crab 2900  cin 3539  Fincfn 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-om 6958  df-er 7629  df-en 7842  df-fin 7845
This theorem is referenced by:  sygbasnfpfi  17755  finptfin  21131  lfinun  21138  nbusgrafi  25977  cusgrasizeindslem2  26003  cusgrasizeinds  26004  hashwwlkext  26274  2spotfi  26419  vdgrfival  26424  vdgrfif  26426  vdusgra0nedg  26435  hashnbgravd  26439  rusgranumwlks  26483  clwlknclwlkdifnum  26488  usgreghash2spot  26596  numclwwlkffin  26609  numclwwlk3lem  26635  numclwwlk4  26637  phpreu  32563  poimirlem25  32604  poimirlem26  32605  poimirlem27  32606  poimirlem28  32607  poimirlem31  32610  poimirlem32  32611  sstotbnd3  32745  hoidmvlelem2  39486  usgrfilem  40546  nbusgrfi  40602  cusgrsizeindslem  40667  cusgrsizeinds  40668  vtxdgfival  40685  vtxdgfisnn0  40690  hashwwlksnext  41120  wwlksnonfi  41127  rusgrnumwwlks  41177  clwwlknclwwlkdifnum  41182  konigsberglem5  41426  fusgreghash2wsp  41502  av-numclwwlkffin  41512  av-numclwwlk3lem  41538  av-numclwwlk4  41540
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