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Theorem tcss 7952
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 4424 . . 3  |-  ( B 
C_  A  ->  B  e.  _V )
3 tcvalg 7946 . . 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 3351 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  x  ->  B  C_  x ) )
65anim1d 559 . . . . 5  |-  ( B 
C_  A  ->  (
( A  C_  x  /\  Tr  x )  -> 
( B  C_  x  /\  Tr  x ) ) )
76ss2abdv 3413 . . . 4  |-  ( B 
C_  A  ->  { x  |  ( A  C_  x  /\  Tr  x ) }  C_  { x  |  ( B  C_  x  /\  Tr  x ) } )
8 intss 4137 . . . 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 7946 . . . 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 3391 . 2  |-  ( B 
C_  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  ( TC `  A ) )
134, 12eqsstrd 3378 1  |-  ( B 
C_  A  ->  ( TC `  B )  C_  ( TC `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   {cab 2419   _Vcvv 2962    C_ wss 3316   |^|cint 4116   Tr wtr 4373   ` cfv 5406   TCctc 7944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-om 6466  df-recs 6818  df-rdg 6852  df-tc 7945
This theorem is referenced by:  hsmexlem4  8586
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