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Theorem tctr 7470
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 4165 . . . 4  |-  ( A. y  e.  { x  |  ( A  C_  x  /\  Tr  x ) } Tr  y  ->  Tr  |^| { x  |  ( A  C_  x  /\  Tr  x ) } )
2 vex 2825 . . . . . 6  |-  y  e. 
_V
3 sseq2 3234 . . . . . . 7  |-  ( x  =  y  ->  ( A  C_  x  <->  A  C_  y
) )
4 treq 4156 . . . . . . 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 2948 . . . . 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 2646 . . 3  |-  Tr  |^| { x  |  ( A 
C_  x  /\  Tr  x ) }
9 tcvalg 7468 . . . 4  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
10 treq 4156 . . . 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 4161 . . 3  |-  Tr  (/)
14 fvprc 5557 . . . 4  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  (/) )
15 treq 4156 . . . 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 1633    e. wcel 1701   {cab 2302   _Vcvv 2822    C_ wss 3186   (/)c0 3489   |^|cint 3899   Tr wtr 4150   ` cfv 5292   TCctc 7466
This theorem is referenced by:  tc2  7472  tcidm  7476  itunitc1  8091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-recs 6430  df-rdg 6465  df-tc 7467
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