Theorem eldm 4875

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 4873	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)

Syntax hints:   ↔ wb 105  ∃wex 1515   ∈ wcel 2176  Vcvv 2772   class class class wbr 4044  dom cdm 4675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-dm 4685

This theorem is referenced by:  dmi  4893  dmcoss  4948  dmcosseq  4950  dminss  5097  dmsnm  5148  dffun7  5298  dffun8  5299  fnres  5392  fndmdif  5685  reldmtpos  6339  dmtpos  6342  tfrexlem  6420