Theorem brelrn 5897

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)

Hypotheses

Ref	Expression
brelrn.1	⊢ 𝐴 ∈ V
brelrn.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brelrn	⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)

Proof of Theorem brelrn

Step	Hyp	Ref	Expression
1		brelrn.1	. 2 ⊢ 𝐴 ∈ V
2		brelrn.2	. 2 ⊢ 𝐵 ∈ V
3		brelrng 5896	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
4	1, 2, 3	mp3an12 1454	1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3429   class class class wbr 5085  ran crn 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  opelrn  5898  dfco2a  6210  cores  6213  dffun9  6527  funcnv  6567  rntpos  8189  rnttrcl  9643  aceq3lem  10042  axdclem  10441  axdclem2  10442  cotr2g  14938  shftfval  15032  psdmrn  18539  metustexhalf  24521  itg1addlem4  25666