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Theorem eldmres 35600
Description: Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 9-Jan-2019.)
Assertion
Ref Expression
eldmres (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem eldmres
StepHypRef Expression
1 eldmg 5754 . 2 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ ∃𝑦 𝐵(𝑅𝐴)𝑦))
2 brres 5847 . . . . 5 (𝑦 ∈ V → (𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴𝐵𝑅𝑦)))
32elv 3485 . . . 4 (𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴𝐵𝑅𝑦))
43exbii 1849 . . 3 (∃𝑦 𝐵(𝑅𝐴)𝑦 ↔ ∃𝑦(𝐵𝐴𝐵𝑅𝑦))
5 19.42v 1955 . . 3 (∃𝑦(𝐵𝐴𝐵𝑅𝑦) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))
64, 5bitri 278 . 2 (∃𝑦 𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))
71, 6syl6bb 290 1 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  wcel 2115  Vcvv 3480   class class class wbr 5052  dom cdm 5542  cres 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-xp 5548  df-dm 5552  df-res 5554
This theorem is referenced by:  eldmres2  35602
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