Theorem eldm 5808

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 5806	. 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 1786   ∈ wcel 2110  Vcvv 3431   class class class wbr 5079  dom cdm 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-dm 5600

This theorem is referenced by:  dmi  5829  dmep  5831  dmcoss  5879  dmcosseq  5881  dminss  6055  dmsnn0  6109  dffun7  6459  dffun8  6460  fnres  6557  opabiota  6848  fndmdif  6916  dff3  6973  frxp  7958  suppvalbr  7972  reldmtpos  8041  dmtpos  8045  aceq3lem  9877  axdc2lem  10205  axdclem2  10277  fpwwe2lem11  10398  nqerf  10687  shftdm  14780  bcthlem4  24489  dchrisumlem3  26637  eulerpath  28601  fundmpss  33736  elfix  34201  fnsingle  34217  fnimage  34227  funpartlem  34240  dfrecs2  34248  dfrdg4  34249  knoppcnlem9  34677  prtlem16  36879  undmrnresiss  41182