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Theorem rntrclfvOAI 35533
Description: The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)
Assertion
Ref Expression
rntrclfvOAI  |-  ( R  e.  V  ->  ran  ( t+ `  R )  =  ran  R )

Proof of Theorem rntrclfvOAI
StepHypRef Expression
1 trclfvub 13071 . . . 4  |-  ( R  e.  V  ->  (
t+ `  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
2 rnss 5063 . . . 4  |-  ( ( t+ `  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) )  ->  ran  ( t+ `  R )  C_  ran  ( R  u.  ( dom  R  X.  ran  R
) ) )
31, 2syl 17 . . 3  |-  ( R  e.  V  ->  ran  ( t+ `  R )  C_  ran  ( R  u.  ( dom  R  X.  ran  R
) ) )
4 rnun 5244 . . . . 5  |-  ran  ( R  u.  ( dom  R  X.  ran  R ) )  =  ( ran 
R  u.  ran  ( dom  R  X.  ran  R
) )
54a1i 11 . . . 4  |-  ( R  e.  V  ->  ran  ( R  u.  ( dom  R  X.  ran  R
) )  =  ( ran  R  u.  ran  ( dom  R  X.  ran  R ) ) )
6 rnxpss 5269 . . . . 5  |-  ran  ( dom  R  X.  ran  R
)  C_  ran  R
7 ssequn2 3607 . . . . 5  |-  ( ran  ( dom  R  X.  ran  R )  C_  ran  R  <-> 
( ran  R  u.  ran  ( dom  R  X.  ran  R ) )  =  ran  R )
86, 7mpbi 212 . . . 4  |-  ( ran 
R  u.  ran  ( dom  R  X.  ran  R
) )  =  ran  R
95, 8syl6eq 2501 . . 3  |-  ( R  e.  V  ->  ran  ( R  u.  ( dom  R  X.  ran  R
) )  =  ran  R )
103, 9sseqtrd 3468 . 2  |-  ( R  e.  V  ->  ran  ( t+ `  R )  C_  ran  R )
11 trclfvlb 13072 . . 3  |-  ( R  e.  V  ->  R  C_  ( t+ `  R ) )
12 rnss 5063 . . 3  |-  ( R 
C_  ( t+ `  R )  ->  ran  R  C_  ran  ( t+ `  R ) )
1311, 12syl 17 . 2  |-  ( R  e.  V  ->  ran  R 
C_  ran  ( t+ `  R ) )
1410, 13eqssd 3449 1  |-  ( R  e.  V  ->  ran  ( t+ `  R )  =  ran  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887    u. cun 3402    C_ wss 3404    X. cxp 4832   dom cdm 4834   ran crn 4835   ` cfv 5582   t+ctcl 13049
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-sbc 3268  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-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-iota 5546  df-fun 5584  df-fv 5590  df-trcl 13051
This theorem is referenced by: (None)
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