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Theorem eldm4 34581
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 5551 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
2 elecALTV 34577 . . . 4 ((𝐴𝑉𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑅𝐴𝑅𝑦))
32elvd 3419 . . 3 (𝐴𝑉 → (𝑦 ∈ [𝐴]𝑅𝐴𝑅𝑦))
43exbidv 2020 . 2 (𝐴𝑉 → (∃𝑦 𝑦 ∈ [𝐴]𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
51, 4bitr4d 274 1 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wex 1878  wcel 2164  Vcvv 3414   class class class wbr 4873  dom cdm 5342  [cec 8007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ec 8011
This theorem is referenced by:  eldmres2  34582
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