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Theorem tcel 8167
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 8160 . 2  |-  ( B  e.  A  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
2 ssel 3483 . . . . . . . 8  |-  ( A 
C_  x  ->  ( B  e.  A  ->  B  e.  x ) )
3 trss 4541 . . . . . . . . 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 429 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  B  C_  x ) )
7 simpr 459 . . . . . . 7  |-  ( ( A  C_  x  /\  Tr  x )  ->  Tr  x )
87a1i 11 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  Tr  x ) )
96, 8jcad 531 . . . . 5  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  -> 
( B  C_  x  /\  Tr  x ) ) )
109ss2abdv 3559 . . . 4  |-  ( B  e.  A  ->  { x  |  ( A  C_  x  /\  Tr  x ) }  C_  { x  |  ( B  C_  x  /\  Tr  x ) } )
11 intss 4292 . . . 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 16 . . 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 8160 . . . 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 3536 . 2  |-  ( B  e.  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  ( TC `  A ) )
171, 16eqsstrd 3523 1  |-  ( B  e.  A  ->  ( TC `  B )  C_  ( TC `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   _Vcvv 3106    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:  tcrank  8293  hsmexlem4  8800
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