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Theorem 2ndrn 6719
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 1st2nd 6717 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 simpr 461 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  R )
31, 2eqeltrrd 2538 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  R )
4 fvex 5796 . . 3  |-  ( 1st `  A )  e.  _V
5 fvex 5796 . . 3  |-  ( 2nd `  A )  e.  _V
64, 5opelrn 5166 . 2  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  R  ->  ( 2nd `  A
)  e.  ran  R
)
73, 6syl 16 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 2nd `  A )  e. 
ran  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   <.cop 3978   ran crn 4936   Rel wrel 4940   ` cfv 5513   1stc1st 6672   2ndc2nd 6673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-iota 5476  df-fun 5515  df-fv 5521  df-1st 6674  df-2nd 6675
This theorem is referenced by:  heicant  28561  mblfinlem1  28563
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