Theorem eldm 5798

Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)

Hypothesis

Ref	Expression
eldm.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
eldm	⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵

Proof of Theorem eldm

Step	Hyp	Ref	Expression
1		eldm.1	. 2 ⊢ 𝐴 ∈ V
2		eldmg 5796	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)

Syntax hints:   ↔ wb 205  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  dom cdm 5580

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

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-dm 5590

This theorem is referenced by:  dmi  5819  dmep  5821  dmcoss  5869  dmcosseq  5871  dminss  6045  dmsnn0  6099  dffun7  6445  dffun8  6446  fnres  6543  opabiota  6833  fndmdif  6901  dff3  6958  frxp  7938  suppvalbr  7952  reldmtpos  8021  dmtpos  8025  aceq3lem  9807  axdc2lem  10135  axdclem2  10207  fpwwe2lem11  10328  nqerf  10617  shftdm  14710  bcthlem4  24396  dchrisumlem3  26544  eulerpath  28506  fundmpss  33646  elfix  34132  fnsingle  34148  fnimage  34158  funpartlem  34171  dfrecs2  34179  dfrdg4  34180  knoppcnlem9  34608  prtlem16  36810  undmrnresiss  41101