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Theorem tcel 7676
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 7669 . 2  |-  ( B  e.  A  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
2 ssel 3334 . . . . . . . 8  |-  ( A 
C_  x  ->  ( B  e.  A  ->  B  e.  x ) )
3 trss 4303 . . . . . . . . 9  |-  ( Tr  x  ->  ( B  e.  x  ->  B  C_  x ) )
43com12 29 . . . . . . . 8  |-  ( B  e.  x  ->  ( Tr  x  ->  B  C_  x ) )
52, 4syl6com 33 . . . . . . 7  |-  ( B  e.  A  ->  ( A  C_  x  ->  ( Tr  x  ->  B  C_  x ) ) )
65imp3a 421 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  B  C_  x ) )
7 simpr 448 . . . . . . 7  |-  ( ( A  C_  x  /\  Tr  x )  ->  Tr  x )
87a1i 11 . . . . . 6  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  ->  Tr  x ) )
96, 8jcad 520 . . . . 5  |-  ( B  e.  A  ->  (
( A  C_  x  /\  Tr  x )  -> 
( B  C_  x  /\  Tr  x ) ) )
109ss2abdv 3408 . . . 4  |-  ( B  e.  A  ->  { x  |  ( A  C_  x  /\  Tr  x ) }  C_  { x  |  ( B  C_  x  /\  Tr  x ) } )
11 intss 4063 . . . 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 7669 . . . 4  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
1513, 14ax-mp 8 . . 3  |-  ( TC
`  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) }
1612, 15syl6sseqr 3387 . 2  |-  ( B  e.  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  ( TC `  A ) )
171, 16eqsstrd 3374 1  |-  ( B  e.  A  ->  ( TC `  B )  C_  ( TC `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    C_ wss 3312   |^|cint 4042   Tr wtr 4294   ` cfv 5446   TCctc 7667
This theorem is referenced by:  tcrank  7800  hsmexlem4  8301
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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