Theorem eldm4 38785

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 5875	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
2		elecALTV 38775	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦))
3	2	elvd 3461	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦))
4	3	exbidv 1942	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦 𝑦 ∈ [𝐴]𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
5	1, 4	bitr4d 284	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅))

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1800   ∈ wcel 2143  Vcvv 3455   class class class wbr 5101  dom cdm 5648  [cec 8676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ec 8680

This theorem is referenced by:  eldmres2  38786