Theorem elfvexd 6917

Description: If a function value has a member, then its argument is a set. Deduction form of elfvex 6916. (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 6916	. 2 ⊢ (𝐴 ∈ (𝐵‘𝐶) → 𝐶 ∈ V)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 3453  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-dm 5671  df-iota 6492  df-fv 6544

This theorem is referenced by:  mrieqv2d  17694  mreexmrid  17698  mreexexlem3d  17701  mreexexlem4d  17702  mreexexd  17703  mreexdomd  17704  acsdomd  18612  ismgmn0  18699  ecqusaddcl  19263  telgsumfz  20059  isirred  20500  tgclb  23106  alexsublem  24180  cnextcn  24203  ustssel  24342  fmucnd  24427  trcfilu  24429  cfiluweak  24430  ucnextcn  24439  imasdsf1olem  24509  imasf1oxmet  24511  comet  24649  restmetu  24706  wlkp1lem4  29990  wlkp1lem8  29994  1wlkdlem4  30457  eupth2lem3lem1  30545  eupth2lem3lem2  30546  gsumsubg  33332  gsummptfzsplitla  33345  opprqusplusg  33737  opprqus0g  33738  lsssra  33944  lbsdiflsp0  33982  fedgmullem1  33985  mzpcl34  43410  xlimbr  46489  xlimmnfvlem2  46495  xlimpnfvlem2  46499  sectpropdlem  49759  invpropdlem  49761  isopropdlem  49763  cicpropdlem  49772  oppcup3  49932  elxpcbasex1ALT  49972  elxpcbasex2ALT  49974  swapf1  49995