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Theorem eldmres2 34381
Description: Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 21-Aug-2020.)
Assertion
Ref Expression
eldmres2 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅   𝑦,𝑉

Proof of Theorem eldmres2
StepHypRef Expression
1 eldmres 34379 . 2 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)))
2 eldmg 5457 . . . 4 (𝐵𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝐵𝑅𝑦))
3 eldm4 34380 . . . 4 (𝐵𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐵]𝑅))
42, 3bitr3d 270 . . 3 (𝐵𝑉 → (∃𝑦 𝐵𝑅𝑦 ↔ ∃𝑦 𝑦 ∈ [𝐵]𝑅))
54anbi2d 614 . 2 (𝐵𝑉 → ((𝐵𝐴 ∧ ∃𝑦 𝐵𝑅𝑦) ↔ (𝐵𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅)))
61, 5bitrd 268 1 (𝐵𝑉 → (𝐵 ∈ dom (𝑅𝐴) ↔ (𝐵𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wex 1852  wcel 2145   class class class wbr 4786  dom cdm 5249  cres 5251  [cec 7894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7898
This theorem is referenced by:  eldmqsres  34394
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