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Theorem tcmin 7672
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 7669 . . . . 5  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
2 fvex 5734 . . . . 5  |-  ( TC
`  A )  e. 
_V
31, 2syl6eqelr 2524 . . . 4  |-  ( A  e.  V  ->  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V )
4 intexab 4350 . . . 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 3555 . . . . . . . . 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 4304 . . . . . . . 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 2951 . . . . . . . . 9  |-  x  e. 
_V
1312inex1 4336 . . . . . . . 8  |-  ( x  i^i  B )  e. 
_V
14 sseq2 3362 . . . . . . . . 9  |-  ( y  =  ( x  i^i 
B )  ->  ( A  C_  y  <->  A  C_  (
x  i^i  B )
) )
15 treq 4300 . . . . . . . . 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 3074 . . . . . . 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 4057 . . . . . . 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 3554 . . . . . 6  |-  ( x  i^i  B )  C_  B
2119, 20syl6ss 3352 . . . . 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 1646 . . 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 7669 . . 3  |-  ( A  e.  V  ->  ( TC `  A )  = 
|^| { y  |  ( A  C_  y  /\  Tr  y ) } )
2625sseq1d 3367 . 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    i^i cin 3311    C_ wss 3312   |^|cint 4042   Tr wtr 4294   ` cfv 5446   TCctc 7667
This theorem is referenced by:  tcidm  7677  tc0  7678  tcwf  7799  itunitc  8293  grur1  8687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-tc 7668
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