Theorem fvclex 7289

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 7251	. . 3 ⊢ ran 𝐹 ∈ V
3		p0ex 4985	. . 3 ⊢ {∅} ∈ V
4	2, 3	unex 7107	. 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V
5		fvclss 6646	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})
6	4, 5	ssexi 4938	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V

Syntax hints:   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  Vcvv 3351   ∪ cun 3721  ∅c0 4063  {csn 4317  ran crn 5251  ‘cfv 6030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-cnv 5258  df-dm 5260  df-rn 5261  df-iota 5993  df-fv 6038

This theorem is referenced by:  fvresex  7290