Theorem relelrn 5930

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)

Assertion

Ref	Expression
relelrn	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)

Proof of Theorem relelrn

Step	Hyp	Ref	Expression
1		brrelex1 5712	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5713	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		brelrng 5926	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3464   class class class wbr 5124  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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  relelrnb  5932  relelrni  5934  dirge  18618  metideq  33929  ntrneinex  44076