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Theorem tcel 8077
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 8070 . 2  |-  ( B  e.  A  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
2 ssel 3459 . . . . . . . 8  |-  ( A 
C_  x  ->  ( B  e.  A  ->  B  e.  x ) )
3 trss 4503 . . . . . . . . 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 431 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  B  C_  x ) )
7 simpr 461 . . . . . . 7  |-  ( ( A  C_  x  /\  Tr  x )  ->  Tr  x )
87a1i 11 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  Tr  x ) )
96, 8jcad 533 . . . . 5  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  -> 
( B  C_  x  /\  Tr  x ) ) )
109ss2abdv 3534 . . . 4  |-  ( B  e.  A  ->  { x  |  ( A  C_  x  /\  Tr  x ) }  C_  { x  |  ( B  C_  x  /\  Tr  x ) } )
11 intss 4258 . . . 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 8070 . . . 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 3512 . 2  |-  ( B  e.  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  ( TC `  A ) )
171, 16eqsstrd 3499 1  |-  ( B  e.  A  ->  ( TC `  B )  C_  ( TC `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   _Vcvv 3078    C_ wss 3437   |^|cint 4237   Tr wtr 4494   ` cfv 5527   TCctc 8068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-recs 6943  df-rdg 6977  df-tc 8069
This theorem is referenced by:  tcrank  8203  hsmexlem4  8710
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