Theorem relelrni 5847

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)

Hypothesis

Ref	Expression
releldm.1	⊢ Rel 𝑅

Assertion

Ref	Expression
relelrni	⊢ (𝐴𝑅𝐵 → 𝐵 ∈ ran 𝑅)

Proof of Theorem relelrni

Step	Hyp	Ref	Expression
1		releldm.1	. 2 ⊢ Rel 𝑅
2		relelrn 5843	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
3	1, 2	mpan 686	1 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ ran 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5070  ran crn 5581  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  fpwwe2lem11  10328  lern  18224  brres2  36334  brfvrcld2  41189