Theorem eldm4 38230

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 5923	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝐴𝑅𝑦))
2		elecALTV 38222	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦))
3	2	elvd 3494	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦))
4	3	exbidv 1920	. 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 1777   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  dom cdm 5700  [cec 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765

This theorem is referenced by:  eldmres2  38231