Theorem fvclex 7775

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 7733	. . 3 ⊢ ran 𝐹 ∈ V
3		snex 5349	. . 3 ⊢ {∅} ∈ V
4	2, 3	unex 7574	. 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V
5		fvclss 7097	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})
6	4, 5	ssexi 5241	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V

Syntax hints:   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  Vcvv 3422   ∪ cun 3881  ∅c0 4253  {csn 4558  ran crn 5581  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-cnv 5588  df-dm 5590  df-rn 5591  df-iota 6376  df-fv 6426

This theorem is referenced by:  fvresex  7776