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Theorem opelrn 5101
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
StepHypRef Expression
1 df-br 4213 . 2  |-  ( A C B  <->  <. A ,  B >.  e.  C )
2 brelrn.1 . . 3  |-  A  e. 
_V
3 brelrn.2 . . 3  |-  B  e. 
_V
42, 3brelrn 5100 . 2  |-  ( A C B  ->  B  e.  ran  C )
51, 4sylbir 205 1  |-  ( <. A ,  B >.  e.  C  ->  B  e.  ran  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   _Vcvv 2956   <.cop 3817   class class class wbr 4212   ran crn 4879
This theorem is referenced by:  zfrep6  5968  2ndrn  6395  disjen  7264  r0weon  7894  gsum2d  15546  dfres3  25382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889
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