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Theorem dmtrclfv 13061
Description: The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
dmtrclfv  |-  ( R  e.  V  ->  dom  ( t+ `  R )  =  dom  R )

Proof of Theorem dmtrclfv
StepHypRef Expression
1 trclfvub 13050 . . . 4  |-  ( R  e.  V  ->  (
t+ `  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
2 dmss 5054 . . . 4  |-  ( ( t+ `  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) )  ->  dom  ( t+ `  R )  C_  dom  ( R  u.  ( dom  R  X.  ran  R
) ) )
31, 2syl 17 . . 3  |-  ( R  e.  V  ->  dom  ( t+ `  R )  C_  dom  ( R  u.  ( dom  R  X.  ran  R
) ) )
4 dmun 5061 . . . 4  |-  dom  ( R  u.  ( dom  R  X.  ran  R ) )  =  ( dom 
R  u.  dom  ( dom  R  X.  ran  R
) )
5 dm0rn0 5071 . . . . . . 7  |-  ( dom 
R  =  (/)  <->  ran  R  =  (/) )
6 xpeq1 4868 . . . . . . . . . 10  |-  ( dom 
R  =  (/)  ->  ( dom  R  X.  ran  R
)  =  ( (/)  X. 
ran  R ) )
7 0xp 4935 . . . . . . . . . 10  |-  ( (/)  X. 
ran  R )  =  (/)
86, 7syl6eq 2486 . . . . . . . . 9  |-  ( dom 
R  =  (/)  ->  ( dom  R  X.  ran  R
)  =  (/) )
98dmeqd 5057 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  (/) )
10 dm0 5068 . . . . . . . . 9  |-  dom  (/)  =  (/)
1110a1i 11 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  dom  (/)  =  (/) )
12 eqcom 2438 . . . . . . . . 9  |-  ( dom 
R  =  (/)  <->  (/)  =  dom  R )
1312biimpi 197 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  (/)  =  dom  R )
149, 11, 133eqtrd 2474 . . . . . . 7  |-  ( dom 
R  =  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  R
)
155, 14sylbir 216 . . . . . 6  |-  ( ran 
R  =  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  R
)
16 dmxp 5073 . . . . . 6  |-  ( ran 
R  =/=  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  R
)
1715, 16pm2.61ine 2744 . . . . 5  |-  dom  ( dom  R  X.  ran  R
)  =  dom  R
1817uneq2i 3623 . . . 4  |-  ( dom 
R  u.  dom  ( dom  R  X.  ran  R
) )  =  ( dom  R  u.  dom  R )
19 unidm 3615 . . . 4  |-  ( dom 
R  u.  dom  R
)  =  dom  R
204, 18, 193eqtri 2462 . . 3  |-  dom  ( R  u.  ( dom  R  X.  ran  R ) )  =  dom  R
213, 20syl6sseq 3516 . 2  |-  ( R  e.  V  ->  dom  ( t+ `  R )  C_  dom  R )
22 trclfvlb 13051 . . 3  |-  ( R  e.  V  ->  R  C_  ( t+ `  R ) )
23 dmss 5054 . . 3  |-  ( R 
C_  ( t+ `  R )  ->  dom  R  C_  dom  ( t+ `  R ) )
2422, 23syl 17 . 2  |-  ( R  e.  V  ->  dom  R 
C_  dom  ( t+ `  R ) )
2521, 24eqssd 3487 1  |-  ( R  e.  V  ->  dom  ( t+ `  R )  =  dom  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    u. cun 3440    C_ wss 3442   (/)c0 3767    X. cxp 4852   dom cdm 4854   ran crn 4855   ` cfv 5601   t+ctcl 13028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-trcl 13030
This theorem is referenced by:  rntrclfvRP  35962
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