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Theorem tcvalg 8061
Description: Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 7947; see tz9.1 8052.) (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tcvalg  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem tcvalg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fveq2 5791 . . 3  |-  ( y  =  A  ->  ( TC `  y )  =  ( TC `  A
) )
2 sseq1 3477 . . . . . 6  |-  ( y  =  A  ->  (
y  C_  x  <->  A  C_  x
) )
32anbi1d 704 . . . . 5  |-  ( y  =  A  ->  (
( y  C_  x  /\  Tr  x )  <->  ( A  C_  x  /\  Tr  x
) ) )
43abbidv 2587 . . . 4  |-  ( y  =  A  ->  { x  |  ( y  C_  x  /\  Tr  x ) }  =  { x  |  ( A  C_  x  /\  Tr  x ) } )
54inteqd 4233 . . 3  |-  ( y  =  A  ->  |^| { x  |  ( y  C_  x  /\  Tr  x ) }  =  |^| { x  |  ( A  C_  x  /\  Tr  x ) } )
61, 5eqeq12d 2473 . 2  |-  ( y  =  A  ->  (
( TC `  y
)  =  |^| { x  |  ( y  C_  x  /\  Tr  x ) }  <->  ( TC `  A )  =  |^| { x  |  ( A 
C_  x  /\  Tr  x ) } ) )
7 vex 3073 . . 3  |-  y  e. 
_V
87tz9.1c 8053 . . 3  |-  |^| { x  |  ( y  C_  x  /\  Tr  x ) }  e.  _V
9 df-tc 8060 . . . 4  |-  TC  =  ( y  e.  _V  |->  |^|
{ x  |  ( y  C_  x  /\  Tr  x ) } )
109fvmpt2 5882 . . 3  |-  ( ( y  e.  _V  /\  |^|
{ x  |  ( y  C_  x  /\  Tr  x ) }  e.  _V )  ->  ( TC
`  y )  = 
|^| { x  |  ( y  C_  x  /\  Tr  x ) } )
117, 8, 10mp2an 672 . 2  |-  ( TC
`  y )  = 
|^| { x  |  ( y  C_  x  /\  Tr  x ) }
126, 11vtoclg 3128 1  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2436   _Vcvv 3070    C_ wss 3428   |^|cint 4228   Tr wtr 4485   ` cfv 5518   TCctc 8059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-om 6579  df-recs 6934  df-rdg 6968  df-tc 8060
This theorem is referenced by:  tcid  8062  tctr  8063  tcmin  8064  tc2  8065  tcsni  8066  tcss  8067  tcel  8068  tcrank  8194
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