Theorem trclublem 13734

Description: If a relation exists then the class of transitive relations which are supersets of that relation is not empty. (Contributed by RP, 28-Apr-2020.)

Assertion

Ref	Expression
trclublem	|- ( R e. V -> ( R u. ( dom R X. ran R ) ) e. { x | ( R C_ x /\ ( x o. x ) C_ x ) } )

Distinct variable group:   x, R

Allowed substitution hint:   V( x)

Proof of Theorem trclublem

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trclexlem 13733	. 2 |- ( R e. V -> ( R u. ( dom R X. ran R ) ) e. _V )
2		ssun1 3776	. . 3 |- R C_ ( R u. ( dom R X. ran R ) )
3		relcnv 5503	. . . . . . . . . . . . . 14 |- Rel `' R
4		relssdmrn 5656	. . . . . . . . . . . . . 14 |- ( Rel `' R -> `' R C_ ( dom `' R X. ran `' R ) )
5	3, 4	ax-mp 5	. . . . . . . . . . . . 13 |- `' R C_ ( dom `' R X. ran `' R )
6		ssequn1 3783	. . . . . . . . . . . . 13 |- ( `' R C_ ( dom `' R X. ran `' R ) <-> ( `' R u. ( dom `' R X. ran `' R ) ) = ( dom `' R X. ran `' R ) )
7	5, 6	mpbi 220	. . . . . . . . . . . 12 |- ( `' R u. ( dom `' R X. ran `' R ) ) = ( dom `' R X. ran `' R )
8		cnvun 5538	. . . . . . . . . . . . 13 |- `' ( R u. ( dom R X. ran R ) ) = ( `' R u. `' ( dom R X. ran R ) )
9		cnvxp 5551	. . . . . . . . . . . . . . 15 |- `' ( dom R X. ran R ) = ( ran R X. dom R )
10		df-rn 5125	. . . . . . . . . . . . . . . 16 |- ran R = dom `' R
11		dfdm4 5316	. . . . . . . . . . . . . . . 16 |- dom R = ran `' R
12	10, 11	xpeq12i 5137	. . . . . . . . . . . . . . 15 |- ( ran R X. dom R ) = ( dom `' R X. ran `' R )
13	9, 12	eqtri 2644	. . . . . . . . . . . . . 14 |- `' ( dom R X. ran R ) = ( dom `' R X. ran `' R )
14	13	uneq2i 3764	. . . . . . . . . . . . 13 |- ( `' R u. `' ( dom R X. ran R ) ) = ( `' R u. ( dom `' R X. ran `' R ) )
15	8, 14	eqtri 2644	. . . . . . . . . . . 12 |- `' ( R u. ( dom R X. ran R ) ) = ( `' R u. ( dom `' R X. ran `' R ) )
16	7, 15, 13	3eqtr4i 2654	. . . . . . . . . . 11 |- `' ( R u. ( dom R X. ran R ) ) = `' ( dom R X. ran R )
17	16	breqi 4659	. . . . . . . . . 10 |- ( b `' ( R u. ( dom R X. ran R ) ) a <-> b `' ( dom R X. ran R ) a )
18		vex 3203	. . . . . . . . . . 11 |- b e. _V
19		vex 3203	. . . . . . . . . . 11 |- a e. _V
20	18, 19	brcnv 5305	. . . . . . . . . 10 |- ( b `' ( R u. ( dom R X. ran R ) ) a <-> a ( R u. ( dom R X. ran R ) ) b )
21	18, 19	brcnv 5305	. . . . . . . . . 10 |- ( b `' ( dom R X. ran R ) a <-> a ( dom R X. ran R ) b )
22	17, 20, 21	3bitr3i 290	. . . . . . . . 9 |- ( a ( R u. ( dom R X. ran R ) ) b <-> a ( dom R X. ran R ) b )
23	16	breqi 4659	. . . . . . . . . 10 |- ( c `' ( R u. ( dom R X. ran R ) ) b <-> c `' ( dom R X. ran R ) b )
24		vex 3203	. . . . . . . . . . 11 |- c e. _V
25	24, 18	brcnv 5305	. . . . . . . . . 10 |- ( c `' ( R u. ( dom R X. ran R ) ) b <-> b ( R u. ( dom R X. ran R ) ) c )
26	24, 18	brcnv 5305	. . . . . . . . . 10 |- ( c `' ( dom R X. ran R ) b <-> b ( dom R X. ran R ) c )
27	23, 25, 26	3bitr3i 290	. . . . . . . . 9 |- ( b ( R u. ( dom R X. ran R ) ) c <-> b ( dom R X. ran R ) c )
28	22, 27	anbi12i 733	. . . . . . . 8 |- ( ( a ( R u. ( dom R X. ran R ) ) b /\ b ( R u. ( dom R X. ran R ) ) c ) <-> ( a ( dom R X. ran R ) b /\ b ( dom R X. ran R ) c ) )
29	28	biimpi 206	. . . . . . 7 |- ( ( a ( R u. ( dom R X. ran R ) ) b /\ b ( R u. ( dom R X. ran R ) ) c ) -> ( a ( dom R X. ran R ) b /\ b ( dom R X. ran R ) c ) )
30	29	eximi 1762	. . . . . 6 |- ( E. b ( a ( R u. ( dom R X. ran R ) ) b /\ b ( R u. ( dom R X. ran R ) ) c ) -> E. b ( a ( dom R X. ran R ) b /\ b ( dom R X. ran R ) c ) )
31	30	ssopab2i 5003	. . . . 5 |- { <. a , c >. | E. b ( a ( R u. ( dom R X. ran R ) ) b /\ b ( R u. ( dom R X. ran R ) ) c ) } C_ { <. a , c >. | E. b ( a ( dom R X. ran R ) b /\ b ( dom R X. ran R ) c ) }
32		df-co 5123	. . . . 5 |- ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) = { <. a , c >. | E. b ( a ( R u. ( dom R X. ran R ) ) b /\ b ( R u. ( dom R X. ran R ) ) c ) }
33		df-co 5123	. . . . 5 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) = { <. a , c >. | E. b ( a ( dom R X. ran R ) b /\ b ( dom R X. ran R ) c ) }
34	31, 32, 33	3sstr4i 3644	. . . 4 |- ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( ( dom R X. ran R ) o. ( dom R X. ran R ) )
35		xptrrel 13719	. . . . 5 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( dom R X. ran R )
36		ssun2 3777	. . . . 5 |- ( dom R X. ran R ) C_ ( R u. ( dom R X. ran R ) )
37	35, 36	sstri 3612	. . . 4 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( R u. ( dom R X. ran R ) )
38	34, 37	sstri 3612	. . 3 |- ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) )
39		trcleq2lem 13730	. . . . 5 |- ( x = ( R u. ( dom R X. ran R ) ) -> ( ( R C_ x /\ ( x o. x ) C_ x ) <-> ( R C_ ( R u. ( dom R X. ran R ) ) /\ ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) ) ) )
40	39	elabg 3351	. . . 4 |- ( ( R u. ( dom R X. ran R ) ) e. _V -> ( ( R u. ( dom R X. ran R ) ) e. { x | ( R C_ x /\ ( x o. x ) C_ x ) } <-> ( R C_ ( R u. ( dom R X. ran R ) ) /\ ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) ) ) )
41	40	biimprd 238	. . 3 |- ( ( R u. ( dom R X. ran R ) ) e. _V -> ( ( R C_ ( R u. ( dom R X. ran R ) ) /\ ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) ) -> ( R u. ( dom R X. ran R ) ) e. { x | ( R C_ x /\ ( x o. x ) C_ x ) } ) )
42	2, 38, 41	mp2ani 714	. 2 |- ( ( R u. ( dom R X. ran R ) ) e. _V -> ( R u. ( dom R X. ran R ) ) e. { x | ( R C_ x /\ ( x o. x ) C_ x ) } )
43	1, 42	syl 17	1 |- ( R e. V -> ( R u. ( dom R X. ran R ) ) e. { x | ( R C_ x /\ ( x o. x ) C_ x ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   u. cun 3572   C_ wss 3574   class class class wbr 4653   {copab 4712   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   o. ccom 5118   Rel wrel 5119

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-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:  trclubi  13735  trclubiOLD  13736  trclubgi  13737  trclubgiOLD  13738  trclub  13739  trclubg  13740