Theorem eldm2g 5893

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 5892	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2		df-br 5115	. . 3 ⊢ (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
3	2	exbii 1876	. 2 ⊢ (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵)
4	1, 3	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1807   ∈ wcel 2150  ⟨cop 4600   class class class wbr 5114  dom cdm 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-dm 5675

This theorem is referenced by:  eldm2  5895  opeldmd  5900  dmfco  6981  releldm2  8043  tfrlem9  8375  climcau  15725  caucvgb  15734  lmff  23441  axhcompl-zf  31320  satfdmlem  35818  dfatdmfcoafv2  47940