Theorem fvclex 7962

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 7913	. . 3 ⊢ ran 𝐹 ∈ V
3		snex 5412	. . 3 ⊢ {∅} ∈ V
4	2, 3	unex 7752	. 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V
5		fvclss 7244	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})
6	4, 5	ssexi 5295	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  Vcvv 3457   ∪ cun 3904  ∅c0 4286  {csn 4591  ran crn 5664  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-cnv 5671  df-dm 5673  df-rn 5674  df-iota 6496  df-fv 6548

This theorem is used by:  fvresex  7963