Theorem elfvexd 6959

Description: If a function value has a member, then its argument is a set. Deduction form of elfvex 6958. (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 6958	. 2 ⊢ (𝐴 ∈ (𝐵‘𝐶) → 𝐶 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3488  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5710  df-iota 6525  df-fv 6581

This theorem is referenced by:  mrieqv2d  17697  mreexmrid  17701  mreexexlem3d  17704  mreexexlem4d  17705  mreexexd  17706  mreexdomd  17707  acsdomd  18627  ismgmn0  18680  ecqusaddcl  19233  telgsumfz  20032  isirred  20445  tgclb  22998  alexsublem  24073  cnextcn  24096  ustssel  24235  fmucnd  24322  trcfilu  24324  cfiluweak  24325  ucnextcn  24334  imasdsf1olem  24404  imasf1oxmet  24406  comet  24547  restmetu  24604  wlkp1lem4  29712  wlkp1lem8  29716  1wlkdlem4  30172  eupth2lem3lem1  30260  eupth2lem3lem2  30261  gsumsubg  33029  opprqusplusg  33482  opprqus0g  33483  lsssra  33603  lbsdiflsp0  33639  fedgmullem1  33642  mzpcl34  42687  xlimbr  45748  xlimmnfvlem2  45754  xlimpnfvlem2  45758