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Theorem fvclex 7031
 Description: Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)
Hypothesis
Ref Expression
fvclex.1 𝐹 ∈ V
Assertion
Ref Expression
fvclex {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V
Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem fvclex
StepHypRef Expression
1 fvclex.1 . . . 4 𝐹 ∈ V
21rnex 6992 . . 3 ran 𝐹 ∈ V
3 p0ex 4779 . . 3 {∅} ∈ V
42, 3unex 6854 . 2 (ran 𝐹 ∪ {∅}) ∈ V
5 fvclss 6404 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
64, 5ssexi 4731 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ran crn 5039  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812 This theorem is referenced by:  fvresex  7032
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