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Theorem unielrel 5538
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 5114 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
2 simpr 461 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  R )
3 vex 3112 . . . . . 6  |-  x  e. 
_V
4 vex 3112 . . . . . 6  |-  y  e. 
_V
53, 4uniopel 4760 . . . . 5  |-  ( <.
x ,  y >.  e.  R  ->  U. <. x ,  y >.  e.  U. R )
65a1i 11 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( <. x ,  y >.  e.  R  ->  U. <. x ,  y
>.  e.  U. R ) )
7 eleq1 2529 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( A  e.  R  <->  <. x ,  y
>.  e.  R ) )
8 unieq 4259 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  U. A  =  U. <. x ,  y >.
)
98eleq1d 2526 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( U. A  e.  U. R  <->  U. <. x ,  y >.  e.  U. R ) )
106, 7, 93imtr4d 268 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( A  e.  R  ->  U. A  e. 
U. R ) )
1110exlimivv 1724 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( A  e.  R  ->  U. A  e. 
U. R ) )
121, 2, 11sylc 60 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  U. A  e.  U. R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   <.cop 4038   U.cuni 4251   Rel wrel 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-opab 4516  df-xp 5014  df-rel 5015
This theorem is referenced by: (None)
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