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Theorem fvclex 7941
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 7899 . . 3 ran 𝐹 ∈ V
3 snex 5430 . . 3 {∅} ∈ V
42, 3unex 7729 . 2 (ran 𝐹 ∪ {∅}) ∈ V
5 fvclss 7237 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
64, 5ssexi 5321 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wex 1781  wcel 2106  {cab 2709  Vcvv 3474  cun 3945  c0 4321  {csn 4627  ran crn 5676  cfv 6540
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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-cnv 5683  df-dm 5685  df-rn 5686  df-iota 6492  df-fv 6548
This theorem is referenced by:  fvresex  7942
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