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Theorem tc2 8190
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 8186 . . . . 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 4559 . . . . . . 7  |-  ( Tr  x  ->  ( A  e.  x  ->  A  C_  x ) )
54imdistanri 691 . . . . . 6  |-  ( ( A  e.  x  /\  Tr  x )  ->  ( A  C_  x  /\  Tr  x ) )
65ss2abi 3568 . . . . 5  |-  { x  |  ( A  e.  x  /\  Tr  x
) }  C_  { x  |  ( A  C_  x  /\  Tr  x ) }
7 intss 4309 . . . . 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 3529 . . 3  |-  ( TC
`  A )  C_  |^|
{ x  |  ( A  e.  x  /\  Tr  x ) }
101elintab 4299 . . . . 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 1623 . . . 4  |-  A  e. 
|^| { x  |  ( A  e.  x  /\  Tr  x ) }
131snss 4156 . . . 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 3675 . 2  |-  ( ( TC `  A )  u.  { A }
)  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
161snid 4060 . . . . 5  |-  A  e. 
{ A }
17 elun2 3668 . . . . 5  |-  ( A  e.  { A }  ->  A  e.  ( ( TC `  A )  u.  { A }
) )
1816, 17ax-mp 5 . . . 4  |-  A  e.  ( ( TC `  A )  u.  { A } )
19 uniun 4270 . . . . . . 7  |-  U. (
( TC `  A
)  u.  { A } )  =  ( U. ( TC `  A )  u.  U. { A } )
20 tctr 8188 . . . . . . . . 9  |-  Tr  ( TC `  A )
21 df-tr 4551 . . . . . . . . 9  |-  ( Tr  ( TC `  A
)  <->  U. ( TC `  A )  C_  ( TC `  A ) )
2220, 21mpbi 208 . . . . . . . 8  |-  U. ( TC `  A )  C_  ( TC `  A )
231unisn 4266 . . . . . . . . 9  |-  U. { A }  =  A
24 tcid 8187 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( TC `  A
) )
251, 24ax-mp 5 . . . . . . . . 9  |-  A  C_  ( TC `  A )
2623, 25eqsstri 3529 . . . . . . . 8  |-  U. { A }  C_  ( TC
`  A )
2722, 26unssi 3675 . . . . . . 7  |-  ( U. ( TC `  A )  u.  U. { A } )  C_  ( TC `  A )
2819, 27eqsstri 3529 . . . . . 6  |-  U. (
( TC `  A
)  u.  { A } )  C_  ( TC `  A )
29 ssun1 3663 . . . . . 6  |-  ( TC
`  A )  C_  ( ( TC `  A )  u.  { A } )
3028, 29sstri 3508 . . . . 5  |-  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } )
31 df-tr 4551 . . . . 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 5882 . . . . . 6  |-  ( TC
`  A )  e. 
_V
34 snex 4697 . . . . . 6  |-  { A }  e.  _V
3533, 34unex 6597 . . . . 5  |-  ( ( TC `  A )  u.  { A }
)  e.  _V
36 eleq2 2530 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( A  e.  x  <->  A  e.  (
( TC `  A
)  u.  { A } ) ) )
37 treq 4556 . . . . . 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 3246 . . . 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 920 . . 3  |-  ( ( TC `  A )  u.  { A }
)  e.  { x  |  ( A  e.  x  /\  Tr  x
) }
41 intss1 4303 . . 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 3515 1  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   _Vcvv 3109    u. cun 3469    C_ wss 3471   {csn 4032   U.cuni 4251   |^|cint 4288   Tr wtr 4550   ` cfv 5594   TCctc 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-tc 8185
This theorem is referenced by:  tcsni  8191
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