Theorem elrn 5791

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 5789	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Syntax hints:   ↔ wb 205  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  dmcosseq  5871  inisegn0  5995  rnco  6145  dffo4  6961  fvclss  7097  rntpos  8026  fpwwe2lem10  10327  fpwwe2lem11  10328  fclim  15190  perfdvf  24972  dftr6  33624  dffr5  33627  brsset  34118  dfon3  34121  brtxpsd  34123  dffix2  34134  elsingles  34147  dfrdg4  34180  undmrnresiss  41101