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Theorem dmtrclfv 13083
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 13072 . . . 4  |-  ( R  e.  V  ->  (
t+ `  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
2 dmss 5053 . . . 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 5060 . . . 4  |-  dom  ( R  u.  ( dom  R  X.  ran  R ) )  =  ( dom 
R  u.  dom  ( dom  R  X.  ran  R
) )
5 dm0rn0 5070 . . . . . . 7  |-  ( dom 
R  =  (/)  <->  ran  R  =  (/) )
6 xpeq1 4867 . . . . . . . . . 10  |-  ( dom 
R  =  (/)  ->  ( dom  R  X.  ran  R
)  =  ( (/)  X. 
ran  R ) )
7 0xp 4934 . . . . . . . . . 10  |-  ( (/)  X. 
ran  R )  =  (/)
86, 7syl6eq 2479 . . . . . . . . 9  |-  ( dom 
R  =  (/)  ->  ( dom  R  X.  ran  R
)  =  (/) )
98dmeqd 5056 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  (/) )
10 dm0 5067 . . . . . . . . 9  |-  dom  (/)  =  (/)
1110a1i 11 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  dom  (/)  =  (/) )
12 eqcom 2431 . . . . . . . . 9  |-  ( dom 
R  =  (/)  <->  (/)  =  dom  R )
1312biimpi 197 . . . . . . . 8  |-  ( dom 
R  =  (/)  ->  (/)  =  dom  R )
149, 11, 133eqtrd 2467 . . . . . . 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 5072 . . . . . 6  |-  ( ran 
R  =/=  (/)  ->  dom  ( dom  R  X.  ran  R )  =  dom  R
)
1715, 16pm2.61ine 2733 . . . . 5  |-  dom  ( dom  R  X.  ran  R
)  =  dom  R
1817uneq2i 3617 . . . 4  |-  ( dom 
R  u.  dom  ( dom  R  X.  ran  R
) )  =  ( dom  R  u.  dom  R )
19 unidm 3609 . . . 4  |-  ( dom 
R  u.  dom  R
)  =  dom  R
204, 18, 193eqtri 2455 . . 3  |-  dom  ( R  u.  ( dom  R  X.  ran  R ) )  =  dom  R
213, 20syl6sseq 3510 . 2  |-  ( R  e.  V  ->  dom  ( t+ `  R )  C_  dom  R )
22 trclfvlb 13073 . . 3  |-  ( R  e.  V  ->  R  C_  ( t+ `  R ) )
23 dmss 5053 . . 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 3481 1  |-  ( R  e.  V  ->  dom  ( t+ `  R )  =  dom  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872    u. cun 3434    C_ wss 3436   (/)c0 3761    X. cxp 4851   dom cdm 4853   ran crn 4854   ` cfv 5601   t+ctcl 13050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-trcl 13052
This theorem is referenced by:  rntrclfvRP  36294
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