Theorem releldmb 5281

Description: Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)

Assertion

Ref	Expression
releldmb	⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem releldmb

Step	Hyp	Ref	Expression
1		eldmg 5241	. . 3 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
2	1	ibi 255	. 2 ⊢ (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3		releldm 5279	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
4	3	ex 449	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅))
5	4	exlimdv 1848	. 2 ⊢ (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅))
6	2, 5	impbid2 215	1 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 195  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583  dom cdm 5038  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048

This theorem is referenced by: (None)