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

Step	Hyp	Ref	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 ) ) )
3	1, 2	syl 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 ) = (/)
8	6, 7	syl6eq 2486	. . . . . . . . 9 |- ( dom R = (/) -> ( dom R X. ran R ) = (/) )
9	8	dmeqd 5057	. . . . . . . 8 |- ( dom R = (/) -> dom ( dom R X. ran R ) = dom (/) )
10		dm0 5068	. . . . . . . . 9 |- dom (/) = (/)
11	10	a1i 11	. . . . . . . 8 |- ( dom R = (/) -> dom (/) = (/) )
12		eqcom 2438	. . . . . . . . 9 |- ( dom R = (/) <-> (/) = dom R )
13	12	biimpi 197	. . . . . . . 8 |- ( dom R = (/) -> (/) = dom R )
14	9, 11, 13	3eqtrd 2474	. . . . . . 7 |- ( dom R = (/) -> dom ( dom R X. ran R ) = dom R )
15	5, 14	sylbir 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 )
17	15, 16	pm2.61ine 2744	. . . . 5 |- dom ( dom R X. ran R ) = dom R
18	17	uneq2i 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
20	4, 18, 19	3eqtri 2462	. . 3 |- dom ( R u. ( dom R X. ran R ) ) = dom R
21	3, 20	syl6sseq 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 ) )
24	22, 23	syl 17	. 2 |- ( R e. V -> dom R C_ dom ( t+ ` R ) )
25	21, 24	eqssd 3487	1 |- ( R e. V -> dom ( t+ ` R ) = dom R )

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