Theorem elrn 5869

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 5867	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Syntax hints:   ↔ wb 208  ∃wex 1799   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100  ran crn 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5655  df-dm 5657  df-rn 5658

This theorem is referenced by:  dmcosseq  5954  dmcosseqOLD  5955  dmcosseqOLDOLD  5956  inisegn0  6087  rnco  6239  rncoOLD  6240  dffo4  7084  fvclss  7225  rntpos  8219  fpwwe2lem10  10598  fpwwe2lem11  10599  fclim  15580  perfdvf  25965  dftr6  36101  dffr5  36104  brsset  36237  dfon3  36240  brtxpsd  36242  dffix2  36253  elsingles  36266  dfrdg4  36301  undmrnresiss  44180