Theorem eldm 4475

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 4473	. 2 ⊢ (A ∈ V → (A ∈ dom B ↔ ∃y ABy))
3	1, 2	ax-mp 7	1 ⊢ (A ∈ dom B ↔ ∃y ABy)

Syntax hints:   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755  dom cdm 4288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298

This theorem is referenced by:  dmi  4493  dmcoss  4544  dmcosseq  4546  dminss  4681  dmsnm  4729  dffun7  4871  dffun8  4872  fnres  4958  fndmdif  5215  reldmtpos  5809  dmtpos  5812  tfrexlem  5889