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Theorem fvclex 7419
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 7381 . . 3 ran 𝐹 ∈ V
3 snex 5142 . . 3 {∅} ∈ V
42, 3unex 7235 . 2 (ran 𝐹 ∪ {∅}) ∈ V
5 fvclss 6774 . 2 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
64, 5ssexi 5042 1 {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wex 1823  wcel 2107  {cab 2763  Vcvv 3398  cun 3790  c0 4141  {csn 4398  ran crn 5358  cfv 6137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-cnv 5365  df-dm 5367  df-rn 5368  df-iota 6101  df-fv 6145
This theorem is referenced by:  fvresex  7420
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