Theorem eldm2g 5838

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 5837	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2		df-br 5090	. . 3 ⊢ (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
3	2	exbii 1849	. 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 1780   ∈ wcel 2111  ⟨cop 4579   class class class wbr 5089  dom cdm 5614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-dm 5624

This theorem is referenced by:  eldm2  5840  opeldmd  5845  dmfco  6918  releldm2  7975  tfrlem9  8304  climcau  15578  caucvgb  15587  lmff  23216  axhcompl-zf  30978  satfdmlem  35412  dfatdmfcoafv2  47364