Theorem eldm4 37609

Description: Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.)

Assertion

Ref	Expression
eldm4	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑉

Proof of Theorem eldm4

Step	Hyp	Ref	Expression
1		eldmg 5898	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
2		elecALTV 37601	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦))
3	2	elvd 3480	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦))
4	3	exbidv 1923	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦 𝑦 ∈ [𝐴]𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
5	1, 4	bitr4d 282	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1780   ∈ wcel 2105  Vcvv 3473   class class class wbr 5148  dom cdm 5676  [cec 8707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8711

This theorem is referenced by:  eldmres2  37610