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Theorem resco 5177
Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
resco  |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )

Proof of Theorem resco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4983 . 2  |-  Rel  (
( A  o.  B
)  |`  C )
2 relco 5171 . 2  |-  Rel  ( A  o.  ( B  |`  C ) )
3 vex 2791 . . . . . 6  |-  x  e. 
_V
4 vex 2791 . . . . . 6  |-  y  e. 
_V
53, 4brco 4852 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
65anbi1i 676 . . . 4  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  ( E. z ( x B z  /\  z A y )  /\  x  e.  C )
)
7 19.41v 1842 . . . 4  |-  ( E. z ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( E. z ( x B z  /\  z A y )  /\  x  e.  C )
)
8 an32 773 . . . . . 6  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
9 vex 2791 . . . . . . . 8  |-  z  e. 
_V
109brres 4961 . . . . . . 7  |-  ( x ( B  |`  C ) z  <->  ( x B z  /\  x  e.  C ) )
1110anbi1i 676 . . . . . 6  |-  ( ( x ( B  |`  C ) z  /\  z A y )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
128, 11bitr4i 243 . . . . 5  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( x
( B  |`  C ) z  /\  z A y ) )
1312exbii 1569 . . . 4  |-  ( E. z ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
146, 7, 133bitr2i 264 . . 3  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
154brres 4961 . . 3  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  ( x ( A  o.  B ) y  /\  x  e.  C ) )
163, 4brco 4852 . . 3  |-  ( x ( A  o.  ( B  |`  C ) ) y  <->  E. z ( x ( B  |`  C ) z  /\  z A y ) )
1714, 15, 163bitr4i 268 . 2  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  x ( A  o.  ( B  |`  C ) ) y )
181, 2, 17eqbrriv 4782 1  |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   class class class wbr 4023    |` cres 4691    o. ccom 4693
This theorem is referenced by:  cocnvcnv2  5184  coires1  5190  relcoi1  5201  dftpos2  6251  canthp1lem2  8275  o1res  12034  gsumzaddlem  15203  tsmsf1o  17827  tsmsmhm  17828  mbfres  18999  hhssims  21852  erdsze2lem2  23735  cvmlift2lem9a  23834  cocnv  26393  diophrw  26838  eldioph2  26841  funcoressn  27990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-co 4698  df-res 4701
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