Theorem releldmb 5927

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 5879	. . 3 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
2	1	ibi 270	. 2 ⊢ (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3		releldm 5925	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
4	3	ex 418	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅))
5	4	exlimdv 1965	. 2 ⊢ (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅))
6	2, 5	impbid2 229	1 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1811   ∈ wcel 2144   class class class wbr 5102  dom cdm 5650  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660

This theorem is used by:  eqvrelref  39720  uobffth  50428  uobeqw  50429  isinito3  50729