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Theorem tcel 8247
Description: The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
Hypothesis
Ref Expression
tc2.1  |-  A  e. 
_V
Assertion
Ref Expression
tcel  |-  ( B  e.  A  ->  ( TC `  B )  C_  ( TC `  A ) )

Proof of Theorem tcel
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tcvalg 8240 . 2  |-  ( B  e.  A  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
2 ssel 3412 . . . . . . . 8  |-  ( A 
C_  x  ->  ( B  e.  A  ->  B  e.  x ) )
3 trss 4499 . . . . . . . . 9  |-  ( Tr  x  ->  ( B  e.  x  ->  B  C_  x ) )
43com12 31 . . . . . . . 8  |-  ( B  e.  x  ->  ( Tr  x  ->  B  C_  x ) )
52, 4syl6com 35 . . . . . . 7  |-  ( B  e.  A  ->  ( A  C_  x  ->  ( Tr  x  ->  B  C_  x ) ) )
65impd 438 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  B  C_  x ) )
7 simpr 468 . . . . . . 7  |-  ( ( A  C_  x  /\  Tr  x )  ->  Tr  x )
87a1i 11 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  Tr  x ) )
96, 8jcad 542 . . . . 5  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  -> 
( B  C_  x  /\  Tr  x ) ) )
109ss2abdv 3488 . . . 4  |-  ( B  e.  A  ->  { x  |  ( A  C_  x  /\  Tr  x ) }  C_  { x  |  ( B  C_  x  /\  Tr  x ) } )
11 intss 4247 . . . 4  |-  ( { x  |  ( A 
C_  x  /\  Tr  x ) }  C_  { x  |  ( B 
C_  x  /\  Tr  x ) }  ->  |^|
{ x  |  ( B  C_  x  /\  Tr  x ) }  C_  |^|
{ x  |  ( A  C_  x  /\  Tr  x ) } )
1210, 11syl 17 . . 3  |-  ( B  e.  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  |^| { x  |  ( A  C_  x  /\  Tr  x ) } )
13 tc2.1 . . . 4  |-  A  e. 
_V
14 tcvalg 8240 . . . 4  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
1513, 14ax-mp 5 . . 3  |-  ( TC
`  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) }
1612, 15syl6sseqr 3465 . 2  |-  ( B  e.  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  ( TC `  A ) )
171, 16eqsstrd 3452 1  |-  ( B  e.  A  ->  ( TC `  B )  C_  ( TC `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   _Vcvv 3031    C_ wss 3390   |^|cint 4226   Tr wtr 4490   ` cfv 5589   TCctc 8238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-tc 8239
This theorem is referenced by:  tcrank  8373  hsmexlem4  8877
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