Theorem eldm4 38590

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 5842	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
2		elecALTV 38580	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦))
3	2	elvd 3433	. . 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 206  ∃wex 1781   ∈ wcel 2114  Vcvv 3427   class class class wbr 5074  dom cdm 5620  [cec 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ec 8634

This theorem is referenced by:  eldmres2  38591