Theorem brelrn 5888

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 5887	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
4	1, 2, 3	mp3an12 1453	1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3437   class class class wbr 5095  ran crn 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-cnv 5629  df-dm 5631  df-rn 5632

This theorem is referenced by:  opelrn  5889  dfco2a  6201  cores  6204  dffun9  6518  funcnv  6558  rntpos  8178  rnttrcl  9623  aceq3lem  10022  axdclem  10421  axdclem2  10422  cotr2g  14890  shftfval  14984  psdmrn  18487  metustexhalf  24491  itg1addlem4  25647