Theorem elfvexd 6869

Description: If a function value has a member, then its argument is a set. Deduction form of elfvex 6868. (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 6868	. 2 ⊢ (𝐴 ∈ (𝐵‘𝐶) → 𝐶 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3439  ‘cfv 6491

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 2707  ax-nul 5250  ax-pr 5376

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-dm 5633  df-iota 6447  df-fv 6499

This theorem is referenced by:  mrieqv2d  17564  mreexmrid  17568  mreexexlem3d  17571  mreexexlem4d  17572  mreexexd  17573  mreexdomd  17574  acsdomd  18482  ismgmn0  18569  ecqusaddcl  19124  telgsumfz  19921  isirred  20357  tgclb  22916  alexsublem  23990  cnextcn  24013  ustssel  24152  fmucnd  24237  trcfilu  24239  cfiluweak  24240  ucnextcn  24249  imasdsf1olem  24319  imasf1oxmet  24321  comet  24459  restmetu  24516  wlkp1lem4  29729  wlkp1lem8  29733  1wlkdlem4  30196  eupth2lem3lem1  30284  eupth2lem3lem2  30285  gsumsubg  33108  gsummptfzsplitla  33121  opprqusplusg  33549  opprqus0g  33550  lsssra  33723  lbsdiflsp0  33762  fedgmullem1  33765  mzpcl34  43010  xlimbr  46108  xlimmnfvlem2  46114  xlimpnfvlem2  46118  sectpropdlem  49318  invpropdlem  49320  isopropdlem  49322  cicpropdlem  49331  oppcup3  49491  elxpcbasex1ALT  49531  elxpcbasex2ALT  49533  swapf1  49554