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Theorem tctr 7425
Description: Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tctr  |-  Tr  ( TC `  A )

Proof of Theorem tctr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trint 4128 . . . 4  |-  ( A. y  e.  { x  |  ( A  C_  x  /\  Tr  x ) } Tr  y  ->  Tr  |^| { x  |  ( A  C_  x  /\  Tr  x ) } )
2 vex 2791 . . . . . 6  |-  y  e. 
_V
3 sseq2 3200 . . . . . . 7  |-  ( x  =  y  ->  ( A  C_  x  <->  A  C_  y
) )
4 treq 4119 . . . . . . 7  |-  ( x  =  y  ->  ( Tr  x  <->  Tr  y )
)
53, 4anbi12d 691 . . . . . 6  |-  ( x  =  y  ->  (
( A  C_  x  /\  Tr  x )  <->  ( A  C_  y  /\  Tr  y
) ) )
62, 5elab 2914 . . . . 5  |-  ( y  e.  { x  |  ( A  C_  x  /\  Tr  x ) }  <-> 
( A  C_  y  /\  Tr  y ) )
76simprbi 450 . . . 4  |-  ( y  e.  { x  |  ( A  C_  x  /\  Tr  x ) }  ->  Tr  y )
81, 7mprg 2612 . . 3  |-  Tr  |^| { x  |  ( A 
C_  x  /\  Tr  x ) }
9 tcvalg 7423 . . . 4  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
10 treq 4119 . . . 4  |-  ( ( TC `  A )  =  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  ->  ( Tr  ( TC `  A )  <->  Tr  |^| { x  |  ( A  C_  x  /\  Tr  x ) } ) )
119, 10syl 15 . . 3  |-  ( A  e.  _V  ->  ( Tr  ( TC `  A
)  <->  Tr  |^| { x  |  ( A  C_  x  /\  Tr  x ) } ) )
128, 11mpbiri 224 . 2  |-  ( A  e.  _V  ->  Tr  ( TC `  A ) )
13 tr0 4124 . . 3  |-  Tr  (/)
14 fvprc 5519 . . . 4  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  (/) )
15 treq 4119 . . . 4  |-  ( ( TC `  A )  =  (/)  ->  ( Tr  ( TC `  A
)  <->  Tr  (/) ) )
1614, 15syl 15 . . 3  |-  ( -.  A  e.  _V  ->  ( Tr  ( TC `  A )  <->  Tr  (/) ) )
1713, 16mpbiri 224 . 2  |-  ( -.  A  e.  _V  ->  Tr  ( TC `  A
) )
1812, 17pm2.61i 156 1  |-  Tr  ( TC `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    C_ wss 3152   (/)c0 3455   |^|cint 3862   Tr wtr 4113   ` cfv 5255   TCctc 7421
This theorem is referenced by:  tc2  7427  tcidm  7431  itunitc1  8046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-tc 7422
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