Theorem brelrn 5356

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 5355	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
4	1, 2, 3	mp3an12 1414	1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653  ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  opelrn  5357  dfco2a  5635  cores  5638  dffun9  5917  funcnv  5958  rntpos  7365  aceq3lem  8943  axdclem  9341  axdclem2  9342  cotr2g  13715  shftfval  13810  psdmrn  17207  metustexhalf  22361  itg1addlem4  23466