Theorem elrn 5836

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 5834	. 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 1779   ∈ wcel 2109  Vcvv 3436   class class class wbr 5092  ran crn 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-cnv 5627  df-dm 5629  df-rn 5630

This theorem is referenced by:  dmcosseq  5919  dmcosseqOLD  5920  dmcosseqOLDOLD  5921  inisegn0  6049  rnco  6201  dffo4  7037  fvclss  7177  rntpos  8172  fpwwe2lem10  10534  fpwwe2lem11  10535  fclim  15460  perfdvf  25802  dftr6  35728  dffr5  35731  brsset  35867  dfon3  35870  brtxpsd  35872  dffix2  35883  elsingles  35896  dfrdg4  35929  undmrnresiss  43581