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Theorem tcmin 7442
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 7439 . . . . 5  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
2 fvex 5555 . . . . 5  |-  ( TC
`  A )  e. 
_V
31, 2syl6eqelr 2385 . . . 4  |-  ( A  e.  V  ->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
4 intexab 4185 . . . 4  |-  ( E. x ( A  C_  x  /\  Tr  x )  <->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
53, 4sylibr 203 . . 3  |-  ( A  e.  V  ->  E. x
( A  C_  x  /\  Tr  x ) )
6 ssin 3404 . . . . . . . . 9  |-  ( ( A  C_  x  /\  A  C_  B )  <->  A  C_  (
x  i^i  B )
)
76biimpi 186 . . . . . . . 8  |-  ( ( A  C_  x  /\  A  C_  B )  ->  A  C_  ( x  i^i 
B ) )
8 trin 4139 . . . . . . . 8  |-  ( ( Tr  x  /\  Tr  B )  ->  Tr  ( x  i^i  B ) )
97, 8anim12i 549 . . . . . . 7  |-  ( ( ( A  C_  x  /\  A  C_  B )  /\  ( Tr  x  /\  Tr  B ) )  ->  ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) ) )
109an4s 799 . . . . . 6  |-  ( ( ( A  C_  x  /\  Tr  x )  /\  ( A  C_  B  /\  Tr  B ) )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) )
1110expcom 424 . . . . 5  |-  ( ( A  C_  B  /\  Tr  B )  ->  (
( A  C_  x  /\  Tr  x )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) ) )
12 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
1312inex1 4171 . . . . . . . 8  |-  ( x  i^i  B )  e. 
_V
14 sseq2 3213 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( A  C_  y  <->  A  C_  (
x  i^i  B )
) )
15 treq 4135 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( Tr  y  <->  Tr  ( x  i^i  B ) ) )
1614, 15anbi12d 691 . . . . . . . 8  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  C_  y  /\  Tr  y )  <->  ( A  C_  ( x  i^i  B
)  /\  Tr  (
x  i^i  B )
) ) )
1713, 16elab 2927 . . . . . . 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 3893 . . . . . . 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 204 . . . . . 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 3403 . . . . . 6  |-  ( x  i^i  B )  C_  B
2119, 20syl6ss 3204 . . . . 5  |-  ( ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B )
2211, 21syl6 29 . . . 4  |-  ( ( A  C_  B  /\  Tr  B )  ->  (
( A  C_  x  /\  Tr  x )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B ) )
2322exlimdv 1626 . . 3  |-  ( ( A  C_  B  /\  Tr  B )  ->  ( E. x ( A  C_  x  /\  Tr  x )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) } 
C_  B ) )
245, 23syl5com 26 . 2  |-  ( A  e.  V  ->  (
( A  C_  B  /\  Tr  B )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B ) )
25 tcvalg 7439 . . 3  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { y  |  ( A  C_  y  /\  Tr  y ) } )
2625sseq1d 3218 . 2  |-  ( A  e.  V  ->  (
( TC `  A
)  C_  B  <->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B )
)
2724, 26sylibrd 225 1  |-  ( A  e.  V  ->  (
( A  C_  B  /\  Tr  B )  -> 
( TC `  A
)  C_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    i^i cin 3164    C_ wss 3165   |^|cint 3878   Tr wtr 4129   ` cfv 5271   TCctc 7437
This theorem is referenced by:  tcidm  7447  tc0  7448  tcwf  7569  itunitc  8063  grur1  8458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-tc 7438
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