Theorem rntrcl 37935

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

Assertion

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

Distinct variable group:   x, X

Allowed substitution hint:   V( x)

Proof of Theorem rntrcl

Step	Hyp	Ref	Expression
1		trclubg 13740	. . . 4 |- ( X e. V -> |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ ( X u. ( dom X X. ran X ) ) )
2		rnss 5354	. . . 4 |- ( |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ ( X u. ( dom X X. ran X ) ) -> ran |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ ran ( X u. ( dom X X. ran X ) ) )
3	1, 2	syl 17	. . 3 |- ( X e. V -> ran |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ ran ( X u. ( dom X X. ran X ) ) )
4		rnun 5541	. . . 4 |- ran ( X u. ( dom X X. ran X ) ) = ( ran X u. ran ( dom X X. ran X ) )
5		rnxpss 5566	. . . . 5 |- ran ( dom X X. ran X ) C_ ran X
6		ssequn2 3786	. . . . 5 |- ( ran ( dom X X. ran X ) C_ ran X <-> ( ran X u. ran ( dom X X. ran X ) ) = ran X )
7	5, 6	mpbi 220	. . . 4 |- ( ran X u. ran ( dom X X. ran X ) ) = ran X
8	4, 7	eqtri 2644	. . 3 |- ran ( X u. ( dom X X. ran X ) ) = ran X
9	3, 8	syl6sseq 3651	. 2 |- ( X e. V -> ran |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } C_ ran X )
10		ssmin 4496	. . 3 |- X C_ |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) }
11		rnss 5354	. . 3 |- ( X C_ |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } -> ran X C_ ran |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } )
12	10, 11	mp1i 13	. 2 |- ( X e. V -> ran X C_ ran |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } )
13	9, 12	eqssd 3620	1 |- ( X e. V -> ran |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } = ran X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   u. cun 3572   C_ wss 3574   |^|cint 4475   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118

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-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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  dfrtrcl5  37936