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Theorem eldm4 35964
 Description: Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.)
Assertion
Ref Expression
eldm4 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑉

Proof of Theorem eldm4
StepHypRef Expression
1 eldmg 5739 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
2 elecALTV 35960 . . . 4 ((𝐴𝑉𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑅𝐴𝑅𝑦))
32elvd 3417 . . 3 (𝐴𝑉 → (𝑦 ∈ [𝐴]𝑅𝐴𝑅𝑦))
43exbidv 1923 . 2 (𝐴𝑉 → (∃𝑦 𝑦 ∈ [𝐴]𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
51, 4bitr4d 285 1 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1782   ∈ wcel 2112  Vcvv 3410   class class class wbr 5033  dom cdm 5525  [cec 8298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ec 8302 This theorem is referenced by:  eldmres2  35965
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