Theorem fvclex 7952

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 = (𝐹‘𝑥)} ∈ V

Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem fvclex

Step	Hyp	Ref	Expression
1		fvclex.1	. . . 4 ⊢ 𝐹 ∈ V
2	1	rnex 7903	. . 3 ⊢ ran 𝐹 ∈ V
3		snex 5408	. . 3 ⊢ {∅} ∈ V
4	2, 3	unex 7739	. 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V
5		fvclss 7237	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})
6	4, 5	ssexi 5290	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V

Syntax hints:   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  Vcvv 3463   ∪ cun 3911  ∅c0 4294  {csn 4591  ran crn 5660  ‘cfv 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6490  df-fv 6542

This theorem is referenced by:  fvresex  7953