Theorem eldm2g 5848

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 5847	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2		df-br 5076	. . 3 ⊢ (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
3	2	exbii 1856	. 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 1787   ∈ wcel 2121  ⟨cop 4564   class class class wbr 5075  dom cdm 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-dm 5631

This theorem is referenced by:  eldm2  5850  opeldmd  5855  dmfco  6927  releldm2  7989  tfrlem9  8318  climcau  15628  caucvgb  15637  lmff  23288  axhcompl-zf  31091  satfdmlem  35611  dfatdmfcoafv2  47731