Theorem eldm 5771

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 5769	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)

Syntax hints:   ↔ wb 208  ∃wex 1780   ∈ wcel 2114  Vcvv 3496   class class class wbr 5068  dom cdm 5557

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-dm 5567

This theorem is referenced by:  dmi  5793  dmep  5795  dmcoss  5844  dmcosseq  5846  dminss  6012  dmsnn0  6066  dffun7  6384  dffun8  6385  fnres  6476  opabiota  6748  fndmdif  6814  dff3  6868  frxp  7822  suppvalbr  7836  reldmtpos  7902  dmtpos  7906  aceq3lem  9548  axdc2lem  9872  axdclem2  9944  fpwwe2lem12  10065  nqerf  10354  shftdm  14432  bcthlem4  23932  dchrisumlem3  26069  eulerpath  28022  fundmpss  33011  elfix  33366  fnsingle  33382  fnimage  33392  funpartlem  33405  dfrecs2  33413  dfrdg4  33414  knoppcnlem9  33842  prtlem16  36007  undmrnresiss  39971