Theorem fvclex 7966

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 7922	. . 3 ⊢ ran 𝐹 ∈ V
3		snex 5435	. . 3 ⊢ {∅} ∈ V
4	2, 3	unex 7752	. 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V
5		fvclss 7255	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})
6	4, 5	ssexi 5324	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V

Syntax hints:   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2704  Vcvv 3471   ∪ cun 3945  ∅c0 4324  {csn 4630  ran crn 5681  ‘cfv 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-cnv 5688  df-dm 5690  df-rn 5691  df-iota 6503  df-fv 6559

This theorem is referenced by:  fvresex  7967