Theorem brelrn 5953

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 5952	. 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 2108  Vcvv 3480   class class class wbr 5143  ran crn 5686

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696

This theorem is referenced by:  opelrn  5954  dfco2a  6266  cores  6269  dffun9  6595  funcnv  6635  rntpos  8264  rnttrcl  9762  aceq3lem  10160  axdclem  10559  axdclem2  10560  cotr2g  15015  shftfval  15109  psdmrn  18618  metustexhalf  24569  itg1addlem4  25734