Theorem opelrn 4993

Description: Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)

Hypotheses

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

Assertion

Ref	Expression
opelrn	|- ( <. A , B >. e. C -> B e. ran C )

Proof of Theorem opelrn

Step	Hyp	Ref	Expression
1		df-br 4112	. 2 |- ( A C B <-> <. A , B >. e. C )
2		brelrn.1	. . 3 |- A e. _V
3		brelrn.2	. . 3 |- B e. _V
4	2, 3	brelrn 4992	. 2 |- ( A C B -> B e. ran C )
5	1, 4	sylbir 135	1 |- ( <. A , B >. e. C -> B e. ran C )

Syntax hints:   -> wi 4   e. wcel 2205   _Vcvv 2815   <.cop 3694   class class class wbr 4111   ran crn 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-cnv 4759  df-dm 4761  df-rn 4762

This theorem is referenced by: (None)