Theorem elrn 5832

Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)

Hypothesis

Ref	Expression
elrn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elrn	⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn

Step	Hyp	Ref	Expression
1		elrn.1	. 2 ⊢ 𝐴 ∈ V
2		elrng 5830	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Syntax hints:   ↔ wb 206  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by:  dmcosseq  5916  dmcosseqOLD  5917  dmcosseqOLDOLD  5918  inisegn0  6046  rnco  6199  rncoOLD  6200  dffo4  7036  fvclss  7175  rntpos  8169  fpwwe2lem10  10531  fpwwe2lem11  10532  fclim  15460  perfdvf  25831  dftr6  35795  dffr5  35798  brsset  35931  dfon3  35934  brtxpsd  35936  dffix2  35947  elsingles  35960  dfrdg4  35995  undmrnresiss  43707