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Theorem 2ndrn 6208
Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
2ndrn  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 2nd `  A )  e. 
ran  R )

Proof of Theorem 2ndrn
StepHypRef Expression
1 simpr 110 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  R )
2 1st2nd 6206 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
32, 1eqeltrrd 2267 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )
4 1stexg 6192 . . . 4  |-  ( A  e.  R  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 6193 . . . 4  |-  ( A  e.  R  ->  ( 2nd `  A )  e. 
_V )
64, 5jca 306 . . 3  |-  ( A  e.  R  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
7 opelrng 4877 . . . 4  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V  /\  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )  ->  ( 2nd `  A
)  e.  ran  R
)
873expa 1205 . . 3  |-  ( ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
96, 8sylan 283 . 2  |-  ( ( A  e.  R  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
101, 3, 9syl2anc 411 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 2nd `  A )  e. 
ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   _Vcvv 2752   <.cop 3610   ran crn 4645   Rel wrel 4649   ` cfv 5235   1stc1st 6163   2ndc2nd 6164
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243  df-1st 6165  df-2nd 6166
This theorem is referenced by: (None)
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