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Theorem rntrcl 36235
Description: The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
Assertion
Ref Expression
rntrcl  |-  ( X  e.  V  ->  ran  |^|
{ x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) }  =  ran  X )
Distinct variable group:    x, X
Allowed substitution hint:    V( x)

Proof of Theorem rntrcl
StepHypRef Expression
1 trclubg 13063 . . . 4  |-  ( X  e.  V  ->  |^| { x  |  ( X  C_  x  /\  ( x  o.  x )  C_  x
) }  C_  ( X  u.  ( dom  X  X.  ran  X ) ) )
2 rnss 5063 . . . 4  |-  ( |^| { x  |  ( X 
C_  x  /\  (
x  o.  x ) 
C_  x ) } 
C_  ( X  u.  ( dom  X  X.  ran  X ) )  ->  ran  |^|
{ x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) }  C_  ran  ( X  u.  ( dom  X  X.  ran  X ) ) )
31, 2syl 17 . . 3  |-  ( X  e.  V  ->  ran  |^|
{ x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) }  C_  ran  ( X  u.  ( dom  X  X.  ran  X ) ) )
4 rnun 5244 . . . 4  |-  ran  ( X  u.  ( dom  X  X.  ran  X ) )  =  ( ran 
X  u.  ran  ( dom  X  X.  ran  X
) )
5 rnxpss 5269 . . . . 5  |-  ran  ( dom  X  X.  ran  X
)  C_  ran  X
6 ssequn2 3607 . . . . 5  |-  ( ran  ( dom  X  X.  ran  X )  C_  ran  X  <-> 
( ran  X  u.  ran  ( dom  X  X.  ran  X ) )  =  ran  X )
75, 6mpbi 212 . . . 4  |-  ( ran 
X  u.  ran  ( dom  X  X.  ran  X
) )  =  ran  X
84, 7eqtri 2473 . . 3  |-  ran  ( X  u.  ( dom  X  X.  ran  X ) )  =  ran  X
93, 8syl6sseq 3478 . 2  |-  ( X  e.  V  ->  ran  |^|
{ x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) }  C_  ran  X )
10 ssmin 4253 . . 3  |-  X  C_  |^|
{ x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) }
11 rnss 5063 . . 3  |-  ( X 
C_  |^| { x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) }  ->  ran  X  C_  ran  |^|
{ x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) } )
1210, 11mp1i 13 . 2  |-  ( X  e.  V  ->  ran  X 
C_  ran  |^| { x  |  ( X  C_  x  /\  ( x  o.  x )  C_  x
) } )
139, 12eqssd 3449 1  |-  ( X  e.  V  ->  ran  |^|
{ x  |  ( X  C_  x  /\  ( x  o.  x
)  C_  x ) }  =  ran  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   {cab 2437    u. cun 3402    C_ wss 3404   |^|cint 4234    X. cxp 4832   dom cdm 4834   ran crn 4835    o. ccom 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846
This theorem is referenced by:  dfrtrcl5  36236
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