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Theorem trclublem 13071
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.
StepHypRef Expression
1 trclexlem 13070 . 2  |-  ( R  e.  V  ->  ( R  u.  ( dom  R  X.  ran  R ) )  e.  _V )
2 ssun1 3599 . . 3  |-  R  C_  ( R  u.  ( dom  R  X.  ran  R
) )
3 relcnv 5210 . . . . . . . . . . . . . 14  |-  Rel  `' R
4 relssdmrn 5359 . . . . . . . . . . . . . 14  |-  ( Rel  `' R  ->  `' R  C_  ( dom  `' R  X.  ran  `' R ) )
53, 4ax-mp 5 . . . . . . . . . . . . 13  |-  `' R  C_  ( dom  `' R  X.  ran  `' R )
6 ssequn1 3606 . . . . . . . . . . . . 13  |-  ( `' R  C_  ( dom  `' R  X.  ran  `' R )  <->  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )  =  ( dom  `' R  X.  ran  `' R ) )
75, 6mpbi 212 . . . . . . . . . . . 12  |-  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )  =  ( dom  `' R  X.  ran  `' R )
8 cnvun 5244 . . . . . . . . . . . . 13  |-  `' ( R  u.  ( dom 
R  X.  ran  R
) )  =  ( `' R  u.  `' ( dom  R  X.  ran  R ) )
9 cnvxp 5257 . . . . . . . . . . . . . . 15  |-  `' ( dom  R  X.  ran  R )  =  ( ran 
R  X.  dom  R
)
10 df-rn 4848 . . . . . . . . . . . . . . . 16  |-  ran  R  =  dom  `' R
11 dfdm4 5030 . . . . . . . . . . . . . . . 16  |-  dom  R  =  ran  `' R
1210, 11xpeq12i 4859 . . . . . . . . . . . . . . 15  |-  ( ran 
R  X.  dom  R
)  =  ( dom  `' R  X.  ran  `' R )
139, 12eqtri 2475 . . . . . . . . . . . . . 14  |-  `' ( dom  R  X.  ran  R )  =  ( dom  `' R  X.  ran  `' R )
1413uneq2i 3587 . . . . . . . . . . . . 13  |-  ( `' R  u.  `' ( dom  R  X.  ran  R ) )  =  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )
158, 14eqtri 2475 . . . . . . . . . . . 12  |-  `' ( R  u.  ( dom 
R  X.  ran  R
) )  =  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )
167, 15, 133eqtr4i 2485 . . . . . . . . . . 11  |-  `' ( R  u.  ( dom 
R  X.  ran  R
) )  =  `' ( dom  R  X.  ran  R )
1716breqi 4411 . . . . . . . . . 10  |-  ( b `' ( R  u.  ( dom  R  X.  ran  R ) ) a  <->  b `' ( dom  R  X.  ran  R ) a )
18 vex 3050 . . . . . . . . . . 11  |-  b  e. 
_V
19 vex 3050 . . . . . . . . . . 11  |-  a  e. 
_V
2018, 19brcnv 5020 . . . . . . . . . 10  |-  ( b `' ( R  u.  ( dom  R  X.  ran  R ) ) a  <->  a ( R  u.  ( dom  R  X.  ran  R ) ) b )
2118, 19brcnv 5020 . . . . . . . . . 10  |-  ( b `' ( dom  R  X.  ran  R ) a  <-> 
a ( dom  R  X.  ran  R ) b )
2217, 20, 213bitr3i 279 . . . . . . . . 9  |-  ( a ( R  u.  ( dom  R  X.  ran  R
) ) b  <->  a ( dom  R  X.  ran  R
) b )
2316breqi 4411 . . . . . . . . . 10  |-  ( c `' ( R  u.  ( dom  R  X.  ran  R ) ) b  <->  c `' ( dom  R  X.  ran  R ) b )
24 vex 3050 . . . . . . . . . . 11  |-  c  e. 
_V
2524, 18brcnv 5020 . . . . . . . . . 10  |-  ( c `' ( R  u.  ( dom  R  X.  ran  R ) ) b  <->  b ( R  u.  ( dom  R  X.  ran  R ) ) c )
2624, 18brcnv 5020 . . . . . . . . . 10  |-  ( c `' ( dom  R  X.  ran  R ) b  <-> 
b ( dom  R  X.  ran  R ) c )
2723, 25, 263bitr3i 279 . . . . . . . . 9  |-  ( b ( R  u.  ( dom  R  X.  ran  R
) ) c  <->  b ( dom  R  X.  ran  R
) c )
2822, 27anbi12i 704 . . . . . . . 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 ) )
2928biimpi 198 . . . . . . 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 ) )
3029eximi 1709 . . . . . 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 ) )
3130ssopab2i 4732 . . . . 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 4846 . . . . 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 4846 . . . . 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 ) }
3431, 32, 333sstr4i 3473 . . . 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 13056 . . . . 5  |-  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) )  C_  ( dom  R  X.  ran  R
)
36 ssun2 3600 . . . . 5  |-  ( dom 
R  X.  ran  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) )
3735, 36sstri 3443 . . . 4  |-  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) )  C_  ( R  u.  ( dom  R  X.  ran  R ) )
3834, 37sstri 3443 . . 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 13067 . . . . 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 ) ) ) ) )
4039elabg 3188 . . . 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 ) ) ) ) )
4140biimprd 227 . . 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
) } ) )
422, 38, 41mp2ani 685 . 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 ) } )
431, 42syl 17 1  |-  ( R  e.  V  ->  ( R  u.  ( dom  R  X.  ran  R ) )  e.  { x  |  ( R  C_  x  /\  ( x  o.  x )  C_  x
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446   E.wex 1665    e. wcel 1889   {cab 2439   _Vcvv 3047    u. cun 3404    C_ wss 3406   class class class wbr 4405   {copab 4463    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    o. ccom 4841   Rel wrel 4842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849
This theorem is referenced by:  trclubi  13072  trclubgi  13073  trclub  13074  trclubg  13075
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