Theorem dmtrcl 36234

Description: The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)

Assertion

Ref	Expression
dmtrcl	|- ( X e. V -> dom |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } = dom X )

Distinct variable group:   x, X

Allowed substitution hint:   V( x)

Proof of Theorem dmtrcl

Step	Hyp	Ref	Expression
1		trclubg 13063	. . . 4 |- ( X e. V -> |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ ( X u. ( dom X X. ran X ) ) )
2		dmss 5034	. . . 4 |- ( |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ ( X u. ( dom X X. ran X ) ) -> dom |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ dom ( X u. ( dom X X. ran X ) ) )
3	1, 2	syl 17	. . 3 |- ( X e. V -> dom |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ dom ( X u. ( dom X X. ran X ) ) )
4		dmun 5041	. . . 4 |- dom ( X u. ( dom X X. ran X ) ) = ( dom X u. dom ( dom X X. ran X ) )
5		dmxpss 5268	. . . . 5 |- dom ( dom X X. ran X ) C_ dom X
6		ssequn2 3607	. . . . 5 |- ( dom ( dom X X. ran X ) C_ dom X <-> ( dom X u. dom ( dom X X. ran X ) ) = dom X )
7	5, 6	mpbi 212	. . . 4 |- ( dom X u. dom ( dom X X. ran X ) ) = dom X
8	4, 7	eqtri 2473	. . 3 |- dom ( X u. ( dom X X. ran X ) ) = dom X
9	3, 8	syl6sseq 3478	. 2 |- ( X e. V -> dom |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ dom X )
10		ssmin 4253	. . 3 |- X C_ |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) }
11		dmss 5034	. . 3 |- ( X C_ |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } -> dom X C_ dom |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } )
12	10, 11	mp1i 13	. 2 |- ( X e. V -> dom X C_ dom |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } )
13	9, 12	eqssd 3449	1 |- ( X e. V -> dom |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } = dom X )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   {cab 2437   u. cun 3402   C_ wss 3404   |^|cint 4234   X. cxp 4832   dom cdm 4834   ran crn 4835   o. ccom 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846

This theorem is referenced by:  dfrtrcl5  36236