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Theorem tc2 7645
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 7641 . . . . 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 4279 . . . . . . 7  |-  ( Tr  x  ->  ( A  e.  x  ->  A  C_  x ) )
54imdistanri 673 . . . . . 6  |-  ( ( A  e.  x  /\  Tr  x )  ->  ( A  C_  x  /\  Tr  x ) )
65ss2abi 3383 . . . . 5  |-  { x  |  ( A  e.  x  /\  Tr  x
) }  C_  { x  |  ( A  C_  x  /\  Tr  x ) }
7 intss 4039 . . . . 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 3346 . . 3  |-  ( TC
`  A )  C_  |^|
{ x  |  ( A  e.  x  /\  Tr  x ) }
101elintab 4029 . . . . 5  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  A. x ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x ) )
11 simpl 444 . . . . 5  |-  ( ( A  e.  x  /\  Tr  x )  ->  A  e.  x )
1210, 11mpgbir 1556 . . . 4  |-  A  e. 
|^| { x  |  ( A  e.  x  /\  Tr  x ) }
131snss 3894 . . . 4  |-  ( A  e.  |^| { x  |  ( A  e.  x  /\  Tr  x ) }  <->  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) } )
1412, 13mpbi 200 . . 3  |-  { A }  C_  |^| { x  |  ( A  e.  x  /\  Tr  x ) }
159, 14unssi 3490 . 2  |-  ( ( TC `  A )  u.  { A }
)  C_  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
161snid 3809 . . . . 5  |-  A  e. 
{ A }
17 elun2 3483 . . . . 5  |-  ( A  e.  { A }  ->  A  e.  ( ( TC `  A )  u.  { A }
) )
1816, 17ax-mp 8 . . . 4  |-  A  e.  ( ( TC `  A )  u.  { A } )
19 uniun 4002 . . . . . . 7  |-  U. (
( TC `  A
)  u.  { A } )  =  ( U. ( TC `  A )  u.  U. { A } )
20 tctr 7643 . . . . . . . . 9  |-  Tr  ( TC `  A )
21 df-tr 4271 . . . . . . . . 9  |-  ( Tr  ( TC `  A
)  <->  U. ( TC `  A )  C_  ( TC `  A ) )
2220, 21mpbi 200 . . . . . . . 8  |-  U. ( TC `  A )  C_  ( TC `  A )
231unisn 3999 . . . . . . . . 9  |-  U. { A }  =  A
24 tcid 7642 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( TC `  A
) )
251, 24ax-mp 8 . . . . . . . . 9  |-  A  C_  ( TC `  A )
2623, 25eqsstri 3346 . . . . . . . 8  |-  U. { A }  C_  ( TC
`  A )
2722, 26unssi 3490 . . . . . . 7  |-  ( U. ( TC `  A )  u.  U. { A } )  C_  ( TC `  A )
2819, 27eqsstri 3346 . . . . . 6  |-  U. (
( TC `  A
)  u.  { A } )  C_  ( TC `  A )
29 ssun1 3478 . . . . . 6  |-  ( TC
`  A )  C_  ( ( TC `  A )  u.  { A } )
3028, 29sstri 3325 . . . . 5  |-  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } )
31 df-tr 4271 . . . . 5  |-  ( Tr  ( ( TC `  A )  u.  { A } )  <->  U. (
( TC `  A
)  u.  { A } )  C_  (
( TC `  A
)  u.  { A } ) )
3230, 31mpbir 201 . . . 4  |-  Tr  (
( TC `  A
)  u.  { A } )
33 fvex 5709 . . . . . 6  |-  ( TC
`  A )  e. 
_V
34 snex 4373 . . . . . 6  |-  { A }  e.  _V
3533, 34unex 4674 . . . . 5  |-  ( ( TC `  A )  u.  { A }
)  e.  _V
36 eleq2 2473 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( A  e.  x  <->  A  e.  (
( TC `  A
)  u.  { A } ) ) )
37 treq 4276 . . . . . 6  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( Tr  x  <->  Tr  ( ( TC `  A )  u.  { A } ) ) )
3836, 37anbi12d 692 . . . . 5  |-  ( x  =  ( ( TC
`  A )  u. 
{ A } )  ->  ( ( A  e.  x  /\  Tr  x )  <->  ( A  e.  ( ( TC `  A )  u.  { A } )  /\  Tr  ( ( TC `  A )  u.  { A } ) ) ) )
3935, 38elab 3050 . . . 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 887 . . 3  |-  ( ( TC `  A )  u.  { A }
)  e.  { x  |  ( A  e.  x  /\  Tr  x
) }
41 intss1 4033 . . 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 3332 1  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2398   _Vcvv 2924    u. cun 3286    C_ wss 3288   {csn 3782   U.cuni 3983   |^|cint 4018   Tr wtr 4270   ` cfv 5421   TCctc 7639
This theorem is referenced by:  tcsni  7646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635  df-tc 7640
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