ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2ndrn Unicode version

Theorem 2ndrn 5888
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 108 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  R )
2 1st2nd 5886 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
32, 1eqeltrrd 2160 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )
4 1stexg 5873 . . . 4  |-  ( A  e.  R  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 5874 . . . 4  |-  ( A  e.  R  ->  ( 2nd `  A )  e. 
_V )
64, 5jca 300 . . 3  |-  ( A  e.  R  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
7 opelrng 4625 . . . 4  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V  /\  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )  ->  ( 2nd `  A
)  e.  ran  R
)
873expa 1139 . . 3  |-  ( ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
96, 8sylan 277 . 2  |-  ( ( A  e.  R  /\  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R
)  ->  ( 2nd `  A )  e.  ran  R )
101, 3, 9syl2anc 403 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 2nd `  A )  e. 
ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   _Vcvv 2612   <.cop 3425   ran crn 4402   Rel wrel 4406   ` cfv 4969   1stc1st 5844   2ndc2nd 5845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fo 4975  df-fv 4977  df-1st 5846  df-2nd 5847
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator