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Theorem eldmres 37774
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 5905 . 2 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ ∃𝑦 𝐵(𝑅𝐴)𝑦))
2 brres 5996 . . . . 5 (𝑦 ∈ V → (𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴𝐵𝑅𝑦)))
32elv 3479 . . . 4 (𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴𝐵𝑅𝑦))
43exbii 1842 . . 3 (∃𝑦 𝐵(𝑅𝐴)𝑦 ↔ ∃𝑦(𝐵𝐴𝐵𝑅𝑦))
5 19.42v 1949 . . 3 (∃𝑦(𝐵𝐴𝐵𝑅𝑦) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))
64, 5bitri 274 . 2 (∃𝑦 𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))
71, 6bitrdi 286 1 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wex 1773  wcel 2098  Vcvv 3473   class class class wbr 5152  dom cdm 5682  cres 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-dm 5692  df-res 5694
This theorem is referenced by:  eldmressnALTV  37776  eldmres2  37779
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