Theorem dmtrclfv 13759

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 13748	. . . 4 |- ( R e. V -> ( t+ ` R ) C_ ( R u. ( dom R X. ran R ) ) )
2		dmss 5323	. . . 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 5331	. . . 4 |- dom ( R u. ( dom R X. ran R ) ) = ( dom R u. dom ( dom R X. ran R ) )
5		dm0rn0 5342	. . . . . . 7 |- ( dom R = (/) <-> ran R = (/) )
6		xpeq1 5128	. . . . . . . . . 10 |- ( dom R = (/) -> ( dom R X. ran R ) = ( (/) X. ran R ) )
7		0xp 5199	. . . . . . . . . 10 |- ( (/) X. ran R ) = (/)
8	6, 7	syl6eq 2672	. . . . . . . . 9 |- ( dom R = (/) -> ( dom R X. ran R ) = (/) )
9	8	dmeqd 5326	. . . . . . . 8 |- ( dom R = (/) -> dom ( dom R X. ran R ) = dom (/) )
10		dm0 5339	. . . . . . . . 9 |- dom (/) = (/)
11	10	a1i 11	. . . . . . . 8 |- ( dom R = (/) -> dom (/) = (/) )
12		eqcom 2629	. . . . . . . . 9 |- ( dom R = (/) <-> (/) = dom R )
13	12	biimpi 206	. . . . . . . 8 |- ( dom R = (/) -> (/) = dom R )
14	9, 11, 13	3eqtrd 2660	. . . . . . 7 |- ( dom R = (/) -> dom ( dom R X. ran R ) = dom R )
15	5, 14	sylbir 225	. . . . . 6 |- ( ran R = (/) -> dom ( dom R X. ran R ) = dom R )
16		dmxp 5344	. . . . . 6 |- ( ran R =/= (/) -> dom ( dom R X. ran R ) = dom R )
17	15, 16	pm2.61ine 2877	. . . . 5 |- dom ( dom R X. ran R ) = dom R
18	17	uneq2i 3764	. . . 4 |- ( dom R u. dom ( dom R X. ran R ) ) = ( dom R u. dom R )
19		unidm 3756	. . . 4 |- ( dom R u. dom R ) = dom R
20	4, 18, 19	3eqtri 2648	. . 3 |- dom ( R u. ( dom R X. ran R ) ) = dom R
21	3, 20	syl6sseq 3651	. 2 |- ( R e. V -> dom ( t+ ` R ) C_ dom R )
22		trclfvlb 13749	. . 3 |- ( R e. V -> R C_ ( t+ ` R ) )
23		dmss 5323	. . 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 3620	1 |- ( R e. V -> dom ( t+ ` R ) = dom R )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   (/)c0 3915   X. cxp 5112   dom cdm 5114   ran crn 5115   ` cfv 5888   t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-trcl 13726

This theorem is referenced by:  rntrclfvRP  38023