Theorem elfvexd 6858

Description: If a function value has a member, then its argument is a set. Deduction form of elfvex 6857. (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 6857	. 2 ⊢ (𝐴 ∈ (𝐵‘𝐶) → 𝐶 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3436  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-dm 5624  df-iota 6437  df-fv 6489

This theorem is referenced by:  mrieqv2d  17545  mreexmrid  17549  mreexexlem3d  17552  mreexexlem4d  17553  mreexexd  17554  mreexdomd  17555  acsdomd  18463  ismgmn0  18550  ecqusaddcl  19105  telgsumfz  19902  isirred  20337  tgclb  22885  alexsublem  23959  cnextcn  23982  ustssel  24121  fmucnd  24206  trcfilu  24208  cfiluweak  24209  ucnextcn  24218  imasdsf1olem  24288  imasf1oxmet  24290  comet  24428  restmetu  24485  wlkp1lem4  29653  wlkp1lem8  29657  1wlkdlem4  30120  eupth2lem3lem1  30208  eupth2lem3lem2  30209  gsumsubg  33026  opprqusplusg  33454  opprqus0g  33455  lsssra  33600  lbsdiflsp0  33639  fedgmullem1  33642  mzpcl34  42823  xlimbr  45924  xlimmnfvlem2  45930  xlimpnfvlem2  45934  sectpropdlem  49136  invpropdlem  49138  isopropdlem  49140  cicpropdlem  49149  oppcup3  49309  elxpcbasex1ALT  49349  elxpcbasex2ALT  49351  swapf1  49372