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Theorem fvclex 7883
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 7841 . . 3 ran 𝐹 ∈ V
3 snex 5386 . . 3 {∅} ∈ V
42, 3unex 7672 . 2 (ran 𝐹 ∪ {∅}) ∈ V
5 fvclss 7185 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
64, 5ssexi 5277 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wex 1781  wcel 2106  {cab 2714  Vcvv 3443  cun 3906  c0 4280  {csn 4584  ran crn 5632  cfv 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-cnv 5639  df-dm 5641  df-rn 5642  df-iota 6445  df-fv 6501
This theorem is referenced by:  fvresex  7884
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