Theorem relelrnb 5935

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

Assertion

Ref	Expression
relelrnb	⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem relelrnb

Step	Hyp	Ref	Expression
1		elrng 5879	. . 3 ⊢ (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
2	1	ibi 270	. 2 ⊢ (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3		relelrn 5933	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
4	3	ex 417	. . 3 ⊢ (Rel 𝑅 → (𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅))
5	4	exlimdv 1960	. 2 ⊢ (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅))
6	2, 5	impbid2 229	1 ⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1806   ∈ wcel 2149   class class class wbr 5110  ran crn 5660  Rel wrel 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  iscard4  44144