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Theorem tc2 8172
Description: A variant of the definition of the transitive closure function, using instead the smallest transitive set containing  A as a member, gives almost the same set, except that  A itself must be added because it is not usually a member of  ( TC `  A
) (and it is never a member if  A is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)
Hypothesis
Ref Expression
tc2.1  |-  A  e. 
_V
Assertion
Ref Expression
tc2  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
Distinct variable group:    x, A

Proof of Theorem tc2
StepHypRef Expression
1 tc2.1 . . . . 5  |-  A  e. 
_V
2 tcvalg 8168 . . . . 5  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
31, 2ax-mp 5 . . . 4  |-  ( TC
`  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) }
4 trss 4549 . . . . . . 7  |-  ( Tr  x  ->  ( A  e.  x  ->  A  C_  x ) )
54imdistanri 691 . . . . . 6  |-  ( ( A  e.  x  /\  Tr  x )  ->  ( A  C_  x  /\  Tr  x ) )
65ss2abi 3572 . . . . 5  |-  { x  |  ( A  e.  x  /\  Tr  x
) }  C_  { x  |  ( A  C_  x  /\  Tr  x ) }
7 intss 4303 . . . . 5  |-  ( { x  |  ( A  e.  x  /\  Tr  x ) }  C_  { x  |  ( A 
C_  x  /\  Tr  x ) }  ->  |^|
{ x  |  ( A  C_  x  /\  Tr  x ) }  C_  |^|
{ x  |  ( A  e.  x  /\  Tr  x ) } )
86, 7ax-mp 5 . . . 4  |-  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
93, 8eqsstri 3534 . . 3  |-  ( TC
`  A )  C_  |^|
{ x  |  ( A  e.  x  /\  Tr  x ) }
101elintab 4293 . . . . 5  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  A. x ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x ) )
11 simpl 457 . . . . 5  |-  ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x )
1210, 11mpgbir 1605 . . . 4  |-  A  e. 
|^| { x  |  ( A  e.  x  /\  Tr  x ) }
131snss 4151 . . . 4  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) } )
1412, 13mpbi 208 . . 3  |-  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x ) }
159, 14unssi 3679 . 2  |-  ( ( TC `  A )  u.  { A }
)  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
161snid 4055 . . . . 5  |-  A  e. 
{ A }
17 elun2 3672 . . . . 5  |-  ( A  e.  { A }  ->  A  e.  ( ( TC `  A )  u.  { A }
) )
1816, 17ax-mp 5 . . . 4  |-  A  e.  ( ( TC `  A )  u.  { A } )
19 uniun 4264 . . . . . . 7  |-  U. (
( TC `  A
)  u.  { A } )  =  ( U. ( TC `  A )  u.  U. { A } )
20 tctr 8170 . . . . . . . . 9  |-  Tr  ( TC `  A )
21 df-tr 4541 . . . . . . . . 9  |-  ( Tr  ( TC `  A
)  <->  U. ( TC `  A )  C_  ( TC `  A ) )
2220, 21mpbi 208 . . . . . . . 8  |-  U. ( TC `  A )  C_  ( TC `  A )
231unisn 4260 . . . . . . . . 9  |-  U. { A }  =  A
24 tcid 8169 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( TC `  A
) )
251, 24ax-mp 5 . . . . . . . . 9  |-  A  C_  ( TC `  A )
2623, 25eqsstri 3534 . . . . . . . 8  |-  U. { A }  C_  ( TC
`  A )
2722, 26unssi 3679 . . . . . . 7  |-  ( U. ( TC `  A )  u.  U. { A } )  C_  ( TC `  A )
2819, 27eqsstri 3534 . . . . . 6  |-  U. (
( TC `  A
)  u.  { A } )  C_  ( TC `  A )
29 ssun1 3667 . . . . . 6  |-  ( TC
`  A )  C_  ( ( TC `  A )  u.  { A } )
3028, 29sstri 3513 . . . . 5  |-  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } )
31 df-tr 4541 . . . . 5  |-  ( Tr  ( ( TC `  A )  u.  { A } )  <->  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } ) )
3230, 31mpbir 209 . . . 4  |-  Tr  (
( TC `  A
)  u.  { A } )
33 fvex 5875 . . . . . 6  |-  ( TC
`  A )  e. 
_V
34 snex 4688 . . . . . 6  |-  { A }  e.  _V
3533, 34unex 6581 . . . . 5  |-  ( ( TC `  A )  u.  { A }
)  e.  _V
36 eleq2 2540 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( A  e.  x  <->  A  e.  (
( TC `  A
)  u.  { A } ) ) )
37 treq 4546 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( Tr  x  <->  Tr  ( ( TC `  A )  u.  { A } ) ) )
3836, 37anbi12d 710 . . . . 5  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( ( A  e.  x  /\  Tr  x )  <->  ( A  e.  ( ( TC `  A )  u.  { A } )  /\  Tr  ( ( TC `  A )  u.  { A } ) ) ) )
3935, 38elab 3250 . . . 4  |-  ( ( ( TC `  A
)  u.  { A } )  e.  {
x  |  ( A  e.  x  /\  Tr  x ) }  <->  ( A  e.  ( ( TC `  A )  u.  { A } )  /\  Tr  ( ( TC `  A )  u.  { A } ) ) )
4018, 32, 39mpbir2an 918 . . 3  |-  ( ( TC `  A )  u.  { A }
)  e.  { x  |  ( A  e.  x  /\  Tr  x
) }
41 intss1 4297 . . 3  |-  ( ( ( TC `  A
)  u.  { A } )  e.  {
x  |  ( A  e.  x  /\  Tr  x ) }  ->  |^|
{ x  |  ( A  e.  x  /\  Tr  x ) }  C_  ( ( TC `  A )  u.  { A } ) )
4240, 41ax-mp 5 . 2  |-  |^| { x  |  ( A  e.  x  /\  Tr  x
) }  C_  (
( TC `  A
)  u.  { A } )
4315, 42eqssi 3520 1  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113    u. cun 3474    C_ wss 3476   {csn 4027   U.cuni 4245   |^|cint 4282   Tr wtr 4540   ` cfv 5587   TCctc 8166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-om 6680  df-recs 7042  df-rdg 7076  df-tc 8167
This theorem is referenced by:  tcsni  8173
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