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Theorem trclubg 12938
Description: The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure (as a relation). (Contributed by RP, 17-May-2020.)
Assertion
Ref Expression
trclubg  |-  ( R  e.  V  ->  |^| { r  |  ( R  C_  r  /\  ( r  o.  r )  C_  r
) }  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
Distinct variable group:    R, r
Allowed substitution hint:    V( r)

Proof of Theorem trclubg
StepHypRef Expression
1 trclublem 12934 . 2  |-  ( R  e.  V  ->  ( R  u.  ( dom  R  X.  ran  R ) )  e.  { r  |  ( R  C_  r  /\  ( r  o.  r )  C_  r
) } )
2 intss1 4303 . 2  |-  ( ( R  u.  ( dom 
R  X.  ran  R
) )  e.  {
r  |  ( R 
C_  r  /\  (
r  o.  r ) 
C_  r ) }  ->  |^| { r  |  ( R  C_  r  /\  ( r  o.  r
)  C_  r ) }  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
31, 2syl 16 1  |-  ( R  e.  V  ->  |^| { r  |  ( R  C_  r  /\  ( r  o.  r )  C_  r
) }  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   {cab 2442    u. cun 3469    C_ wss 3471   |^|cint 4288    X. cxp 5006   dom cdm 5008   ran crn 5009    o. ccom 5012
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-8 1821  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-pow 4634  ax-pr 4695  ax-un 6591
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-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020
This theorem is referenced by: (None)
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