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Theorem tcmin 8067
Description: Defining property of the transitive closure function: it is a subset of any transitive class containing  A. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tcmin  |-  ( A  e.  V  ->  (
( A  C_  B  /\  Tr  B )  -> 
( TC `  A
)  C_  B )
)

Proof of Theorem tcmin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tcvalg 8064 . . . . 5  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
2 fvex 5804 . . . . 5  |-  ( TC
`  A )  e. 
_V
31, 2syl6eqelr 2549 . . . 4  |-  ( A  e.  V  ->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
4 intexab 4553 . . . 4  |-  ( E. x ( A  C_  x  /\  Tr  x )  <->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
53, 4sylibr 212 . . 3  |-  ( A  e.  V  ->  E. x
( A  C_  x  /\  Tr  x ) )
6 ssin 3675 . . . . . . . . 9  |-  ( ( A  C_  x  /\  A  C_  B )  <->  A  C_  (
x  i^i  B )
)
76biimpi 194 . . . . . . . 8  |-  ( ( A  C_  x  /\  A  C_  B )  ->  A  C_  ( x  i^i 
B ) )
8 trin 4498 . . . . . . . 8  |-  ( ( Tr  x  /\  Tr  B )  ->  Tr  ( x  i^i  B ) )
97, 8anim12i 566 . . . . . . 7  |-  ( ( ( A  C_  x  /\  A  C_  B )  /\  ( Tr  x  /\  Tr  B ) )  ->  ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) ) )
109an4s 822 . . . . . 6  |-  ( ( ( A  C_  x  /\  Tr  x )  /\  ( A  C_  B  /\  Tr  B ) )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) )
1110expcom 435 . . . . 5  |-  ( ( A  C_  B  /\  Tr  B )  ->  (
( A  C_  x  /\  Tr  x )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) ) )
12 vex 3075 . . . . . . . . 9  |-  x  e. 
_V
1312inex1 4536 . . . . . . . 8  |-  ( x  i^i  B )  e. 
_V
14 sseq2 3481 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( A  C_  y  <->  A  C_  (
x  i^i  B )
) )
15 treq 4494 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( Tr  y  <->  Tr  ( x  i^i  B ) ) )
1614, 15anbi12d 710 . . . . . . . 8  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  C_  y  /\  Tr  y )  <->  ( A  C_  ( x  i^i  B
)  /\  Tr  (
x  i^i  B )
) ) )
1713, 16elab 3207 . . . . . . 7  |-  ( ( x  i^i  B )  e.  { y  |  ( A  C_  y  /\  Tr  y ) }  <-> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) )
18 intss1 4246 . . . . . . 7  |-  ( ( x  i^i  B )  e.  { y  |  ( A  C_  y  /\  Tr  y ) }  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) } 
C_  ( x  i^i 
B ) )
1917, 18sylbir 213 . . . . . 6  |-  ( ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  ( x  i^i  B ) )
20 inss2 3674 . . . . . 6  |-  ( x  i^i  B )  C_  B
2119, 20syl6ss 3471 . . . . 5  |-  ( ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B )
2211, 21syl6 33 . . . 4  |-  ( ( A  C_  B  /\  Tr  B )  ->  (
( A  C_  x  /\  Tr  x )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B ) )
2322exlimdv 1691 . . 3  |-  ( ( A  C_  B  /\  Tr  B )  ->  ( E. x ( A  C_  x  /\  Tr  x )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) } 
C_  B ) )
245, 23syl5com 30 . 2  |-  ( A  e.  V  ->  (
( A  C_  B  /\  Tr  B )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B ) )
25 tcvalg 8064 . . 3  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { y  |  ( A  C_  y  /\  Tr  y ) } )
2625sseq1d 3486 . 2  |-  ( A  e.  V  ->  (
( TC `  A
)  C_  B  <->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B )
)
2724, 26sylibrd 234 1  |-  ( A  e.  V  ->  (
( A  C_  B  /\  Tr  B )  -> 
( TC `  A
)  C_  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2437   _Vcvv 3072    i^i cin 3430    C_ wss 3431   |^|cint 4231   Tr wtr 4488   ` cfv 5521   TCctc 8062
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-om 6582  df-recs 6937  df-rdg 6971  df-tc 8063
This theorem is referenced by:  tcidm  8072  tc0  8073  tcwf  8196  itunitc  8696  grur1  9093
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