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Theorem tc2 8252
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 8248 . . . . 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 4520 . . . . . . 7  |-  ( Tr  x  ->  ( A  e.  x  ->  A  C_  x ) )
54imdistanri 702 . . . . . 6  |-  ( ( A  e.  x  /\  Tr  x )  ->  ( A  C_  x  /\  Tr  x ) )
65ss2abi 3513 . . . . 5  |-  { x  |  ( A  e.  x  /\  Tr  x
) }  C_  { x  |  ( A  C_  x  /\  Tr  x ) }
7 intss 4269 . . . . 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 3474 . . 3  |-  ( TC
`  A )  C_  |^|
{ x  |  ( A  e.  x  /\  Tr  x ) }
101elintab 4259 . . . . 5  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  A. x ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x ) )
11 simpl 463 . . . . 5  |-  ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x )
1210, 11mpgbir 1684 . . . 4  |-  A  e. 
|^| { x  |  ( A  e.  x  /\  Tr  x ) }
131snss 4109 . . . 4  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) } )
1412, 13mpbi 213 . . 3  |-  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x ) }
159, 14unssi 3621 . 2  |-  ( ( TC `  A )  u.  { A }
)  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
161snid 4008 . . . . 5  |-  A  e. 
{ A }
17 elun2 3614 . . . . 5  |-  ( A  e.  { A }  ->  A  e.  ( ( TC `  A )  u.  { A }
) )
1816, 17ax-mp 5 . . . 4  |-  A  e.  ( ( TC `  A )  u.  { A } )
19 uniun 4231 . . . . . . 7  |-  U. (
( TC `  A
)  u.  { A } )  =  ( U. ( TC `  A )  u.  U. { A } )
20 tctr 8250 . . . . . . . . 9  |-  Tr  ( TC `  A )
21 df-tr 4512 . . . . . . . . 9  |-  ( Tr  ( TC `  A
)  <->  U. ( TC `  A )  C_  ( TC `  A ) )
2220, 21mpbi 213 . . . . . . . 8  |-  U. ( TC `  A )  C_  ( TC `  A )
231unisn 4227 . . . . . . . . 9  |-  U. { A }  =  A
24 tcid 8249 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( TC `  A
) )
251, 24ax-mp 5 . . . . . . . . 9  |-  A  C_  ( TC `  A )
2623, 25eqsstri 3474 . . . . . . . 8  |-  U. { A }  C_  ( TC
`  A )
2722, 26unssi 3621 . . . . . . 7  |-  ( U. ( TC `  A )  u.  U. { A } )  C_  ( TC `  A )
2819, 27eqsstri 3474 . . . . . 6  |-  U. (
( TC `  A
)  u.  { A } )  C_  ( TC `  A )
29 ssun1 3609 . . . . . 6  |-  ( TC
`  A )  C_  ( ( TC `  A )  u.  { A } )
3028, 29sstri 3453 . . . . 5  |-  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } )
31 df-tr 4512 . . . . 5  |-  ( Tr  ( ( TC `  A )  u.  { A } )  <->  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } ) )
3230, 31mpbir 214 . . . 4  |-  Tr  (
( TC `  A
)  u.  { A } )
33 fvex 5898 . . . . . 6  |-  ( TC
`  A )  e. 
_V
34 snex 4655 . . . . . 6  |-  { A }  e.  _V
3533, 34unex 6616 . . . . 5  |-  ( ( TC `  A )  u.  { A }
)  e.  _V
36 eleq2 2529 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( A  e.  x  <->  A  e.  (
( TC `  A
)  u.  { A } ) ) )
37 treq 4517 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( Tr  x  <->  Tr  ( ( TC `  A )  u.  { A } ) ) )
3836, 37anbi12d 722 . . . . 5  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( ( A  e.  x  /\  Tr  x )  <->  ( A  e.  ( ( TC `  A )  u.  { A } )  /\  Tr  ( ( TC `  A )  u.  { A } ) ) ) )
3935, 38elab 3197 . . . 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 936 . . 3  |-  ( ( TC `  A )  u.  { A }
)  e.  { x  |  ( A  e.  x  /\  Tr  x
) }
41 intss1 4263 . . 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 3460 1  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   {cab 2448   _Vcvv 3057    u. cun 3414    C_ wss 3416   {csn 3980   U.cuni 4212   |^|cint 4248   Tr wtr 4511   ` cfv 5601   TCctc 8246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6720  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-tc 8247
This theorem is referenced by:  tcsni  8253
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