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Theorem tcmin 8163
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 8160 . . . . 5  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
2 fvex 5858 . . . . 5  |-  ( TC
`  A )  e. 
_V
31, 2syl6eqelr 2551 . . . 4  |-  ( A  e.  V  ->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
4 intexab 4595 . . . 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 3706 . . . . . . . . 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 4542 . . . . . . . 8  |-  ( ( Tr  x  /\  Tr  B )  ->  Tr  ( x  i^i  B ) )
97, 8anim12i 564 . . . . . . 7  |-  ( ( ( A  C_  x  /\  A  C_  B )  /\  ( Tr  x  /\  Tr  B ) )  ->  ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) ) )
109an4s 824 . . . . . 6  |-  ( ( ( A  C_  x  /\  Tr  x )  /\  ( A  C_  B  /\  Tr  B ) )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) )
1110expcom 433 . . . . 5  |-  ( ( A  C_  B  /\  Tr  B )  ->  (
( A  C_  x  /\  Tr  x )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) ) )
12 vex 3109 . . . . . . . . 9  |-  x  e. 
_V
1312inex1 4578 . . . . . . . 8  |-  ( x  i^i  B )  e. 
_V
14 sseq2 3511 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( A  C_  y  <->  A  C_  (
x  i^i  B )
) )
15 treq 4538 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( Tr  y  <->  Tr  ( x  i^i  B ) ) )
1614, 15anbi12d 708 . . . . . . . 8  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  C_  y  /\  Tr  y )  <->  ( A  C_  ( x  i^i  B
)  /\  Tr  (
x  i^i  B )
) ) )
1713, 16elab 3243 . . . . . . 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 4286 . . . . . . 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 3705 . . . . . 6  |-  ( x  i^i  B )  C_  B
2119, 20syl6ss 3501 . . . . 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 1729 . . 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 8160 . . 3  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { y  |  ( A  C_  y  /\  Tr  y ) } )
2625sseq1d 3516 . 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 367    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   _Vcvv 3106    i^i cin 3460    C_ wss 3461   |^|cint 4271   Tr wtr 4532   ` cfv 5570   TCctc 8158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-tc 8159
This theorem is referenced by:  tcidm  8168  tc0  8169  tcwf  8292  itunitc  8792  grur1  9187
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