Theorem brelrn 3342

Description: The second argument of a binary relation belongs to its range.

Hypotheses

Ref	Expression
brelrn.1	|- A e. V
brelrn.2	|- B e. V

Assertion

Ref	Expression
brelrn	|- (ACB -> B e. ran C)

Proof of Theorem brelrn

Step	Hyp	Ref	Expression
1		brelrn.2	. 2 |- B e. V
2		brelrn.1	. . 3 |- A e. V
3		brelrng 3341	. . 3 |- ((A e. V /\ B e. V /\ ACB) -> B e. ran C)
4	2, 3	mp3an1 903	. 2 |- ((B e. V /\ ACB) -> B e. ran C)
5	1, 4	mpan 695	1 |- (ACB -> B e. ran C)

Syntax hints:   -> wi 3   e. wcel 958  Vcvv 1809   class class class wbr 2617  ran crn 3169

This theorem is referenced by:  opelrn 3343  cores 3497  dffun8 3539  funcnv 3555  cbvfo 3883  psdmrn 8605  rnhmpha 10477

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2701  ax-pow 2740  ax-pr 2777

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2618  df-opab 2665  df-cnv 3184  df-dm 3186  df-rn 3187

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