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

Step	Hyp	Ref	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 ) ) )
3	1, 2	syl 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 ) = (/)
8	6, 7	syl6eq 2479	. . . . . . . . 9 |- ( dom R = (/) -> ( dom R X. ran R ) = (/) )
9	8	dmeqd 5056	. . . . . . . 8 |- ( dom R = (/) -> dom ( dom R X. ran R ) = dom (/) )
10		dm0 5067	. . . . . . . . 9 |- dom (/) = (/)
11	10	a1i 11	. . . . . . . 8 |- ( dom R = (/) -> dom (/) = (/) )
12		eqcom 2431	. . . . . . . . 9 |- ( dom R = (/) <-> (/) = dom R )
13	12	biimpi 197	. . . . . . . 8 |- ( dom R = (/) -> (/) = dom R )
14	9, 11, 13	3eqtrd 2467	. . . . . . 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 5072	. . . . . 6 |- ( ran R =/= (/) -> dom ( dom R X. ran R ) = dom R )
17	15, 16	pm2.61ine 2733	. . . . 5 |- dom ( dom R X. ran R ) = dom R
18	17	uneq2i 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
20	4, 18, 19	3eqtri 2455	. . 3 |- dom ( R u. ( dom R X. ran R ) ) = dom R
21	3, 20	syl6sseq 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 ) )
24	22, 23	syl 17	. 2 |- ( R e. V -> dom R C_ dom ( t+ ` R ) )
25	21, 24	eqssd 3481	1 |- ( R e. V -> dom ( t+ ` R ) = dom R )

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