Theorem elfvexd 6874

Description: If a function value has a member, then its argument is a set. Deduction form of elfvex 6873. (An artifact of our function value definition.) (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
elfvexd.1	⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶))

Assertion

Ref	Expression
elfvexd	⊢ (𝜑 → 𝐶 ∈ V)

Proof of Theorem elfvexd

Step	Hyp	Ref	Expression
1		elfvexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶))
2		elfvex 6873	. 2 ⊢ (𝐴 ∈ (𝐵‘𝐶) → 𝐶 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430  ‘cfv 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-dm 5638  df-iota 6452  df-fv 6504

This theorem is referenced by:  mrieqv2d  17602  mreexmrid  17606  mreexexlem3d  17609  mreexexlem4d  17610  mreexexd  17611  mreexdomd  17612  acsdomd  18520  ismgmn0  18607  ecqusaddcl  19165  telgsumfz  19962  isirred  20396  tgclb  22951  alexsublem  24025  cnextcn  24048  ustssel  24187  fmucnd  24272  trcfilu  24274  cfiluweak  24275  ucnextcn  24284  imasdsf1olem  24354  imasf1oxmet  24356  comet  24494  restmetu  24551  wlkp1lem4  29764  wlkp1lem8  29768  1wlkdlem4  30231  eupth2lem3lem1  30319  eupth2lem3lem2  30320  gsumsubg  33128  gsummptfzsplitla  33141  opprqusplusg  33570  opprqus0g  33571  lsssra  33753  lbsdiflsp0  33792  fedgmullem1  33795  mzpcl34  43185  xlimbr  46281  xlimmnfvlem2  46287  xlimpnfvlem2  46291  sectpropdlem  49531  invpropdlem  49533  isopropdlem  49535  cicpropdlem  49544  oppcup3  49704  elxpcbasex1ALT  49744  elxpcbasex2ALT  49746  swapf1  49767