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Theorem trclublem 13048
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 13047 . 2  |-  ( R  e.  V  ->  ( R  u.  ( dom  R  X.  ran  R ) )  e.  _V )
2 ssun1 3629 . . 3  |-  R  C_  ( R  u.  ( dom  R  X.  ran  R
) )
3 relcnv 5223 . . . . . . . . . . . . . 14  |-  Rel  `' R
4 relssdmrn 5372 . . . . . . . . . . . . . 14  |-  ( Rel  `' R  ->  `' R  C_  ( dom  `' R  X.  ran  `' R ) )
53, 4ax-mp 5 . . . . . . . . . . . . 13  |-  `' R  C_  ( dom  `' R  X.  ran  `' R )
6 ssequn1 3636 . . . . . . . . . . . . 13  |-  ( `' R  C_  ( dom  `' R  X.  ran  `' R )  <->  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )  =  ( dom  `' R  X.  ran  `' R ) )
75, 6mpbi 211 . . . . . . . . . . . 12  |-  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )  =  ( dom  `' R  X.  ran  `' R )
8 cnvun 5257 . . . . . . . . . . . . 13  |-  `' ( R  u.  ( dom 
R  X.  ran  R
) )  =  ( `' R  u.  `' ( dom  R  X.  ran  R ) )
9 cnvxp 5270 . . . . . . . . . . . . . . 15  |-  `' ( dom  R  X.  ran  R )  =  ( ran 
R  X.  dom  R
)
10 df-rn 4861 . . . . . . . . . . . . . . . 16  |-  ran  R  =  dom  `' R
11 dfdm4 5043 . . . . . . . . . . . . . . . 16  |-  dom  R  =  ran  `' R
1210, 11xpeq12i 4872 . . . . . . . . . . . . . . 15  |-  ( ran 
R  X.  dom  R
)  =  ( dom  `' R  X.  ran  `' R )
139, 12eqtri 2451 . . . . . . . . . . . . . 14  |-  `' ( dom  R  X.  ran  R )  =  ( dom  `' R  X.  ran  `' R )
1413uneq2i 3617 . . . . . . . . . . . . 13  |-  ( `' R  u.  `' ( dom  R  X.  ran  R ) )  =  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )
158, 14eqtri 2451 . . . . . . . . . . . 12  |-  `' ( R  u.  ( dom 
R  X.  ran  R
) )  =  ( `' R  u.  ( dom  `' R  X.  ran  `' R ) )
167, 15, 133eqtr4i 2461 . . . . . . . . . . 11  |-  `' ( R  u.  ( dom 
R  X.  ran  R
) )  =  `' ( dom  R  X.  ran  R )
1716breqi 4426 . . . . . . . . . 10  |-  ( b `' ( R  u.  ( dom  R  X.  ran  R ) ) a  <->  b `' ( dom  R  X.  ran  R ) a )
18 vex 3084 . . . . . . . . . . 11  |-  b  e. 
_V
19 vex 3084 . . . . . . . . . . 11  |-  a  e. 
_V
2018, 19brcnv 5033 . . . . . . . . . 10  |-  ( b `' ( R  u.  ( dom  R  X.  ran  R ) ) a  <->  a ( R  u.  ( dom  R  X.  ran  R ) ) b )
2118, 19brcnv 5033 . . . . . . . . . 10  |-  ( b `' ( dom  R  X.  ran  R ) a  <-> 
a ( dom  R  X.  ran  R ) b )
2217, 20, 213bitr3i 278 . . . . . . . . 9  |-  ( a ( R  u.  ( dom  R  X.  ran  R
) ) b  <->  a ( dom  R  X.  ran  R
) b )
2316breqi 4426 . . . . . . . . . 10  |-  ( c `' ( R  u.  ( dom  R  X.  ran  R ) ) b  <->  c `' ( dom  R  X.  ran  R ) b )
24 vex 3084 . . . . . . . . . . 11  |-  c  e. 
_V
2524, 18brcnv 5033 . . . . . . . . . 10  |-  ( c `' ( R  u.  ( dom  R  X.  ran  R ) ) b  <->  b ( R  u.  ( dom  R  X.  ran  R ) ) c )
2624, 18brcnv 5033 . . . . . . . . . 10  |-  ( c `' ( dom  R  X.  ran  R ) b  <-> 
b ( dom  R  X.  ran  R ) c )
2723, 25, 263bitr3i 278 . . . . . . . . 9  |-  ( b ( R  u.  ( dom  R  X.  ran  R
) ) c  <->  b ( dom  R  X.  ran  R
) c )
2822, 27anbi12i 701 . . . . . . . 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 197 . . . . . . 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 1702 . . . . . 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 4745 . . . . 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 4859 . . . . 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 4859 . . . . 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 3503 . . . 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 13033 . . . . 5  |-  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) )  C_  ( dom  R  X.  ran  R
)
36 ssun2 3630 . . . . 5  |-  ( dom 
R  X.  ran  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) )
3735, 36sstri 3473 . . . 4  |-  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) )  C_  ( R  u.  ( dom  R  X.  ran  R ) )
3834, 37sstri 3473 . . 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 13044 . . . . 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 3219 . . . 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 226 . . 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 682 . 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 370    = wceq 1437   E.wex 1659    e. wcel 1868   {cab 2407   _Vcvv 3081    u. cun 3434    C_ wss 3436   class class class wbr 4420   {copab 4478    X. cxp 4848   `'ccnv 4849   dom cdm 4850   ran crn 4851    o. ccom 4854   Rel wrel 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862
This theorem is referenced by:  trclubi  13049  trclubgi  13050  trclub  13051  trclubg  13052
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