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Theorem 2ndrn 6011
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 109 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  R )
2 1st2nd 6009 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
32, 1eqeltrrd 2177 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )
4 1stexg 5996 . . . 4  |-  ( A  e.  R  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 5997 . . . 4  |-  ( A  e.  R  ->  ( 2nd `  A )  e. 
_V )
64, 5jca 302 . . 3  |-  ( A  e.  R  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
7 opelrng 4709 . . . 4  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V  /\  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )  ->  ( 2nd `  A
)  e.  ran  R
)
873expa 1149 . . 3  |-  ( ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
96, 8sylan 279 . 2  |-  ( ( A  e.  R  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
101, 3, 9syl2anc 406 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 2nd `  A )  e. 
ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1448   _Vcvv 2641   <.cop 3477   ran crn 4478   Rel wrel 4482   ` cfv 5059   1stc1st 5967   2ndc2nd 5968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fo 5065  df-fv 5067  df-1st 5969  df-2nd 5970
This theorem is referenced by: (None)
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