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Theorem dmtrclfv 13093
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 13082 . . . 4  |-  ( R  e.  V  ->  (
t+ `  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
2 dmss 5012 . . . 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 5019 . . . 4  |-  dom  ( R  u.  ( dom  R  X.  ran  R ) )  =  ( dom 
R  u.  dom  ( dom  R  X.  ran  R
) )
5 dm0rn0 5029 . . . . . . 7  |-  ( dom 
R  =  (/)  <->  ran  R  =  (/) )
6 xpeq1 4826 . . . . . . . . . 10  |-  ( dom 
R  =  (/)  ->  ( dom  R  X.  ran  R
)  =  ( (/)  X. 
ran  R ) )
7 0xp 4893 . . . . . . . . . 10  |-  ( (/)  X. 
ran  R )  =  (/)
86, 7syl6eq 2502 . . . . . . . . 9  |-  ( dom 
R  =  (/)  ->  ( dom  R  X.  ran  R
)  =  (/) )
98dmeqd 5015 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  (/) )
10 dm0 5026 . . . . . . . . 9  |-  dom  (/)  =  (/)
1110a1i 11 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  dom  (/)  =  (/) )
12 eqcom 2459 . . . . . . . . 9  |-  ( dom 
R  =  (/)  <->  (/)  =  dom  R )
1312biimpi 199 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  (/)  =  dom  R )
149, 11, 133eqtrd 2490 . . . . . . 7  |-  ( dom 
R  =  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  R
)
155, 14sylbir 218 . . . . . 6  |-  ( ran 
R  =  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  R
)
16 dmxp 5031 . . . . . 6  |-  ( ran 
R  =/=  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  R
)
1715, 16pm2.61ine 2707 . . . . 5  |-  dom  ( dom  R  X.  ran  R
)  =  dom  R
1817uneq2i 3553 . . . 4  |-  ( dom 
R  u.  dom  ( dom  R  X.  ran  R
) )  =  ( dom  R  u.  dom  R )
19 unidm 3545 . . . 4  |-  ( dom 
R  u.  dom  R
)  =  dom  R
204, 18, 193eqtri 2478 . . 3  |-  dom  ( R  u.  ( dom  R  X.  ran  R ) )  =  dom  R
213, 20syl6sseq 3446 . 2  |-  ( R  e.  V  ->  dom  ( t+ `  R )  C_  dom  R )
22 trclfvlb 13083 . . 3  |-  ( R  e.  V  ->  R  C_  ( t+ `  R ) )
23 dmss 5012 . . 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 3417 1  |-  ( R  e.  V  ->  dom  ( t+ `  R )  =  dom  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1448    e. wcel 1891    u. cun 3370    C_ wss 3372   (/)c0 3699    X. cxp 4810   dom cdm 4812   ran crn 4813   ` cfv 5561   t+ctcl 13060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-int 4205  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-iota 5525  df-fun 5563  df-fv 5569  df-trcl 13062
This theorem is referenced by:  rntrclfvRP  36325
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