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Theorem eldmres 34360
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 5474 . 2 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ ∃𝑦 𝐵(𝑅𝐴)𝑦))
2 brresALTV 34356 . . . . 5 (𝑦 ∈ V → (𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴𝐵𝑅𝑦)))
32elv 34309 . . . 4 (𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴𝐵𝑅𝑦))
43exbii 1923 . . 3 (∃𝑦 𝐵(𝑅𝐴)𝑦 ↔ ∃𝑦(𝐵𝐴𝐵𝑅𝑦))
5 19.42v 2030 . . 3 (∃𝑦(𝐵𝐴𝐵𝑅𝑦) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))
64, 5bitri 264 . 2 (∃𝑦 𝐵(𝑅𝐴)𝑦 ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))
71, 6syl6bb 276 1 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1853  wcel 2139  Vcvv 3340   class class class wbr 4804  dom cdm 5266  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-dm 5276  df-res 5278
This theorem is referenced by:  eldmres2  34362
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