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Theorem tc2 7517
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 7513 . . . . 5  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
31, 2ax-mp 8 . . . 4  |-  ( TC
`  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) }
4 trss 4203 . . . . . . 7  |-  ( Tr  x  ->  ( A  e.  x  ->  A  C_  x ) )
54imdistanri 672 . . . . . 6  |-  ( ( A  e.  x  /\  Tr  x )  ->  ( A  C_  x  /\  Tr  x ) )
65ss2abi 3321 . . . . 5  |-  { x  |  ( A  e.  x  /\  Tr  x
) }  C_  { x  |  ( A  C_  x  /\  Tr  x ) }
7 intss 3964 . . . . 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 8 . . . 4  |-  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
93, 8eqsstri 3284 . . 3  |-  ( TC
`  A )  C_  |^|
{ x  |  ( A  e.  x  /\  Tr  x ) }
101elintab 3954 . . . . 5  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  A. x ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x ) )
11 simpl 443 . . . . 5  |-  ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x )
1210, 11mpgbir 1550 . . . 4  |-  A  e. 
|^| { x  |  ( A  e.  x  /\  Tr  x ) }
131snss 3824 . . . 4  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) } )
1412, 13mpbi 199 . . 3  |-  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x ) }
159, 14unssi 3426 . 2  |-  ( ( TC `  A )  u.  { A }
)  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
161snid 3743 . . . . 5  |-  A  e. 
{ A }
17 elun2 3419 . . . . 5  |-  ( A  e.  { A }  ->  A  e.  ( ( TC `  A )  u.  { A }
) )
1816, 17ax-mp 8 . . . 4  |-  A  e.  ( ( TC `  A )  u.  { A } )
19 uniun 3927 . . . . . . 7  |-  U. (
( TC `  A
)  u.  { A } )  =  ( U. ( TC `  A )  u.  U. { A } )
20 tctr 7515 . . . . . . . . 9  |-  Tr  ( TC `  A )
21 df-tr 4195 . . . . . . . . 9  |-  ( Tr  ( TC `  A
)  <->  U. ( TC `  A )  C_  ( TC `  A ) )
2220, 21mpbi 199 . . . . . . . 8  |-  U. ( TC `  A )  C_  ( TC `  A )
231unisn 3924 . . . . . . . . 9  |-  U. { A }  =  A
24 tcid 7514 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( TC `  A
) )
251, 24ax-mp 8 . . . . . . . . 9  |-  A  C_  ( TC `  A )
2623, 25eqsstri 3284 . . . . . . . 8  |-  U. { A }  C_  ( TC
`  A )
2722, 26unssi 3426 . . . . . . 7  |-  ( U. ( TC `  A )  u.  U. { A } )  C_  ( TC `  A )
2819, 27eqsstri 3284 . . . . . 6  |-  U. (
( TC `  A
)  u.  { A } )  C_  ( TC `  A )
29 ssun1 3414 . . . . . 6  |-  ( TC
`  A )  C_  ( ( TC `  A )  u.  { A } )
3028, 29sstri 3264 . . . . 5  |-  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } )
31 df-tr 4195 . . . . 5  |-  ( Tr  ( ( TC `  A )  u.  { A } )  <->  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } ) )
3230, 31mpbir 200 . . . 4  |-  Tr  (
( TC `  A
)  u.  { A } )
33 fvex 5622 . . . . . 6  |-  ( TC
`  A )  e. 
_V
34 snex 4297 . . . . . 6  |-  { A }  e.  _V
3533, 34unex 4600 . . . . 5  |-  ( ( TC `  A )  u.  { A }
)  e.  _V
36 eleq2 2419 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( A  e.  x  <->  A  e.  (
( TC `  A
)  u.  { A } ) ) )
37 treq 4200 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( Tr  x  <->  Tr  ( ( TC `  A )  u.  { A } ) ) )
3836, 37anbi12d 691 . . . . 5  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( ( A  e.  x  /\  Tr  x )  <->  ( A  e.  ( ( TC `  A )  u.  { A } )  /\  Tr  ( ( TC `  A )  u.  { A } ) ) ) )
3935, 38elab 2990 . . . 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 886 . . 3  |-  ( ( TC `  A )  u.  { A }
)  e.  { x  |  ( A  e.  x  /\  Tr  x
) }
41 intss1 3958 . . 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 8 . 2  |-  |^| { x  |  ( A  e.  x  /\  Tr  x
) }  C_  (
( TC `  A
)  u.  { A } )
4315, 42eqssi 3271 1  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   _Vcvv 2864    u. cun 3226    C_ wss 3228   {csn 3716   U.cuni 3908   |^|cint 3943   Tr wtr 4194   ` cfv 5337   TCctc 7511
This theorem is referenced by:  tcsni  7518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-tc 7512
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