Theorem eldm2g 5877

Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
eldm2g	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem eldm2g

Step	Hyp	Ref	Expression
1		eldmg 5876	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2		df-br 5103	. . 3 ⊢ (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
3	2	exbii 1870	. 2 ⊢ (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵)
4	1, 3	bitrdi 289	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1801   ∈ wcel 2144  ⟨cop 4590   class class class wbr 5102  dom cdm 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-dm 5659

This theorem is referenced by:  eldm2  5879  opeldmd  5884  dmfco  6965  releldm2  8026  tfrlem9  8358  climcau  15700  caucvgb  15709  lmff  23363  axhcompl-zf  31203  satfdmlem  35723  dfatdmfcoafv2  47853