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Theorem tcss 8070
Description: The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
Hypothesis
Ref Expression
tc2.1  |-  A  e. 
_V
Assertion
Ref Expression
tcss  |-  ( B 
C_  A  ->  ( TC `  B )  C_  ( TC `  A ) )

Proof of Theorem tcss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tc2.1 . . . 4  |-  A  e. 
_V
21ssex 4539 . . 3  |-  ( B 
C_  A  ->  B  e.  _V )
3 tcvalg 8064 . . 3  |-  ( B  e.  _V  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
42, 3syl 16 . 2  |-  ( B 
C_  A  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
5 sstr2 3466 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  x  ->  B  C_  x ) )
65anim1d 564 . . . . 5  |-  ( B 
C_  A  ->  (
( A  C_  x  /\  Tr  x )  -> 
( B  C_  x  /\  Tr  x ) ) )
76ss2abdv 3528 . . . 4  |-  ( B 
C_  A  ->  { x  |  ( A  C_  x  /\  Tr  x ) }  C_  { x  |  ( B  C_  x  /\  Tr  x ) } )
8 intss 4252 . . . 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 ) } )
97, 8syl 16 . . 3  |-  ( B 
C_  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  |^| { x  |  ( A  C_  x  /\  Tr  x ) } )
10 tcvalg 8064 . . . 4  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
111, 10ax-mp 5 . . 3  |-  ( TC
`  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) }
129, 11syl6sseqr 3506 . 2  |-  ( B 
C_  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  ( TC `  A ) )
134, 12eqsstrd 3493 1  |-  ( B 
C_  A  ->  ( TC `  B )  C_  ( TC `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2437   _Vcvv 3072    C_ wss 3431   |^|cint 4231   Tr wtr 4488   ` cfv 5521   TCctc 8062
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-om 6582  df-recs 6937  df-rdg 6971  df-tc 8063
This theorem is referenced by:  hsmexlem4  8704
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