Theorem eldm2g 5913

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 5912	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2		df-br 5149	. . 3 ⊢ (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
3	2	exbii 1845	. 2 ⊢ (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵)
4	1, 3	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1776   ∈ wcel 2106  ⟨cop 4637   class class class wbr 5148  dom cdm 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699

This theorem is referenced by:  eldm2  5915  opeldmd  5920  dmfco  7005  releldm2  8067  tfrlem9  8424  climcau  15704  caucvgb  15713  lmff  23325  axhcompl-zf  31027  satfdmlem  35353  dfatdmfcoafv2  47204