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Theorem 2ndrn 6088
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 6086 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
32, 1eqeltrrd 2218 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )
4 1stexg 6072 . . . 4  |-  ( A  e.  R  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 6073 . . . 4  |-  ( A  e.  R  ->  ( 2nd `  A )  e. 
_V )
64, 5jca 304 . . 3  |-  ( A  e.  R  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
7 opelrng 4778 . . . 4  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V  /\  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )  ->  ( 2nd `  A
)  e.  ran  R
)
873expa 1182 . . 3  |-  ( ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
96, 8sylan 281 . 2  |-  ( ( A  e.  R  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
101, 3, 9syl2anc 409 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 1481   _Vcvv 2689   <.cop 3534   ran crn 4547   Rel wrel 4551   ` cfv 5130   1stc1st 6043   2ndc2nd 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fo 5136  df-fv 5138  df-1st 6045  df-2nd 6046
This theorem is referenced by: (None)
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