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Theorem unielrel 5197
Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
unielrel  |-  ( ( Rel  R  /\  A  e.  R )  ->  U. A  e.  U. R )

Proof of Theorem unielrel
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrel 4789 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
2 simpr 447 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  R )
3 vex 2791 . . . . . 6  |-  x  e. 
_V
4 vex 2791 . . . . . 6  |-  y  e. 
_V
53, 4uniopel 4270 . . . . 5  |-  ( <.
x ,  y >.  e.  R  ->  U. <. x ,  y >.  e.  U. R )
65a1i 10 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( <. x ,  y >.  e.  R  ->  U. <. x ,  y
>.  e.  U. R ) )
7 eleq1 2343 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( A  e.  R  <->  <. x ,  y
>.  e.  R ) )
8 unieq 3836 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  U. A  =  U. <. x ,  y >.
)
98eleq1d 2349 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( U. A  e.  U. R  <->  U. <. x ,  y >.  e.  U. R ) )
106, 7, 93imtr4d 259 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( A  e.  R  ->  U. A  e. 
U. R ) )
1110exlimivv 1667 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( A  e.  R  ->  U. A  e. 
U. R ) )
121, 2, 11sylc 56 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  U. A  e.  U. R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643   U.cuni 3827   Rel wrel 4694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-opab 4078  df-xp 4695  df-rel 4696
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