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Theorem tcss 7445
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 4174 . . 3  |-  ( B 
C_  A  ->  B  e.  _V )
3 tcvalg 7439 . . 3  |-  ( B  e.  _V  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
42, 3syl 15 . 2  |-  ( B 
C_  A  ->  ( TC `  B )  = 
|^| { x  |  ( B  C_  x  /\  Tr  x ) } )
5 sstr2 3199 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  x  ->  B  C_  x ) )
65anim1d 547 . . . . 5  |-  ( B 
C_  A  ->  (
( A  C_  x  /\  Tr  x )  -> 
( B  C_  x  /\  Tr  x ) ) )
76ss2abdv 3259 . . . 4  |-  ( B 
C_  A  ->  { x  |  ( A  C_  x  /\  Tr  x ) }  C_  { x  |  ( B  C_  x  /\  Tr  x ) } )
8 intss 3899 . . . 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 15 . . 3  |-  ( B 
C_  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  |^| { x  |  ( A  C_  x  /\  Tr  x ) } )
10 tcvalg 7439 . . . 4  |-  ( A  e.  _V  ->  ( TC `  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) } )
111, 10ax-mp 8 . . 3  |-  ( TC
`  A )  = 
|^| { x  |  ( A  C_  x  /\  Tr  x ) }
129, 11syl6sseqr 3238 . 2  |-  ( B 
C_  A  ->  |^| { x  |  ( B  C_  x  /\  Tr  x ) }  C_  ( TC `  A ) )
134, 12eqsstrd 3225 1  |-  ( B 
C_  A  ->  ( TC `  B )  C_  ( TC `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165   |^|cint 3878   Tr wtr 4129   ` cfv 5271   TCctc 7437
This theorem is referenced by:  hsmexlem4  8071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-tc 7438
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