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Theorem tcmin 7614
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 7611 . . . . 5  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
2 fvex 5683 . . . . 5  |-  ( TC
`  A )  e. 
_V
31, 2syl6eqelr 2477 . . . 4  |-  ( A  e.  V  ->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
4 intexab 4300 . . . 4  |-  ( E. x ( A  C_  x  /\  Tr  x )  <->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
53, 4sylibr 204 . . 3  |-  ( A  e.  V  ->  E. x
( A  C_  x  /\  Tr  x ) )
6 ssin 3507 . . . . . . . . 9  |-  ( ( A  C_  x  /\  A  C_  B )  <->  A  C_  (
x  i^i  B )
)
76biimpi 187 . . . . . . . 8  |-  ( ( A  C_  x  /\  A  C_  B )  ->  A  C_  ( x  i^i 
B ) )
8 trin 4254 . . . . . . . 8  |-  ( ( Tr  x  /\  Tr  B )  ->  Tr  ( x  i^i  B ) )
97, 8anim12i 550 . . . . . . 7  |-  ( ( ( A  C_  x  /\  A  C_  B )  /\  ( Tr  x  /\  Tr  B ) )  ->  ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) ) )
109an4s 800 . . . . . 6  |-  ( ( ( A  C_  x  /\  Tr  x )  /\  ( A  C_  B  /\  Tr  B ) )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) )
1110expcom 425 . . . . 5  |-  ( ( A  C_  B  /\  Tr  B )  ->  (
( A  C_  x  /\  Tr  x )  -> 
( A  C_  (
x  i^i  B )  /\  Tr  ( x  i^i 
B ) ) ) )
12 vex 2903 . . . . . . . . 9  |-  x  e. 
_V
1312inex1 4286 . . . . . . . 8  |-  ( x  i^i  B )  e. 
_V
14 sseq2 3314 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( A  C_  y  <->  A  C_  (
x  i^i  B )
) )
15 treq 4250 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( Tr  y  <->  Tr  ( x  i^i  B ) ) )
1614, 15anbi12d 692 . . . . . . . 8  |-  ( y  =  ( x  i^i 
B )  ->  (
( A  C_  y  /\  Tr  y )  <->  ( A  C_  ( x  i^i  B
)  /\  Tr  (
x  i^i  B )
) ) )
1713, 16elab 3026 . . . . . . 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 4008 . . . . . . 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 205 . . . . . 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 3506 . . . . . 6  |-  ( x  i^i  B )  C_  B
2119, 20syl6ss 3304 . . . . 5  |-  ( ( A  C_  ( x  i^i  B )  /\  Tr  ( x  i^i  B ) )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B )
2211, 21syl6 31 . . . 4  |-  ( ( A  C_  B  /\  Tr  B )  ->  (
( A  C_  x  /\  Tr  x )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B ) )
2322exlimdv 1643 . . 3  |-  ( ( A  C_  B  /\  Tr  B )  ->  ( E. x ( A  C_  x  /\  Tr  x )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) } 
C_  B ) )
245, 23syl5com 28 . 2  |-  ( A  e.  V  ->  (
( A  C_  B  /\  Tr  B )  ->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B ) )
25 tcvalg 7611 . . 3  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { y  |  ( A  C_  y  /\  Tr  y ) } )
2625sseq1d 3319 . 2  |-  ( A  e.  V  ->  (
( TC `  A
)  C_  B  <->  |^| { y  |  ( A  C_  y  /\  Tr  y ) }  C_  B )
)
2724, 26sylibrd 226 1  |-  ( A  e.  V  ->  (
( A  C_  B  /\  Tr  B )  -> 
( TC `  A
)  C_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   _Vcvv 2900    i^i cin 3263    C_ wss 3264   |^|cint 3993   Tr wtr 4244   ` cfv 5395   TCctc 7609
This theorem is referenced by:  tcidm  7619  tc0  7620  tcwf  7741  itunitc  8235  grur1  8629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605  df-tc 7610
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