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Theorem resco 5346
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 5138 . 2  |-  Rel  (
( A  o.  B
)  |`  C )
2 relco 5340 . 2  |-  Rel  ( A  o.  ( B  |`  C ) )
3 vex 3034 . . . . . 6  |-  x  e. 
_V
4 vex 3034 . . . . . 6  |-  y  e. 
_V
53, 4brco 5010 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
65anbi1i 709 . . . 4  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  ( E. z ( x B z  /\  z A y )  /\  x  e.  C )
)
7 19.41v 1838 . . . 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 815 . . . . . 6  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
9 vex 3034 . . . . . . . 8  |-  z  e. 
_V
109brres 5117 . . . . . . 7  |-  ( x ( B  |`  C ) z  <->  ( x B z  /\  x  e.  C ) )
1110anbi1i 709 . . . . . 6  |-  ( ( x ( B  |`  C ) z  /\  z A y )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
128, 11bitr4i 260 . . . . 5  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( x
( B  |`  C ) z  /\  z A y ) )
1312exbii 1726 . . . 4  |-  ( E. z ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
146, 7, 133bitr2i 281 . . 3  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
154brres 5117 . . 3  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  ( x ( A  o.  B ) y  /\  x  e.  C ) )
163, 4brco 5010 . . 3  |-  ( x ( A  o.  ( B  |`  C ) ) y  <->  E. z ( x ( B  |`  C ) z  /\  z A y ) )
1714, 15, 163bitr4i 285 . 2  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  x ( A  o.  ( B  |`  C ) ) y )
181, 2, 17eqbrriv 4935 1  |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   class class class wbr 4395    |` cres 4841    o. ccom 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-co 4848  df-res 4851
This theorem is referenced by:  cocnvcnv2  5354  coires1  5360  relcoi1OLD  5372  dftpos2  7008  canthp1lem2  9096  o1res  13701  gsumzaddlem  17632  tsmsf1o  21237  tsmsmhm  21238  mbfres  22679  hhssims  27007  erdsze2lem2  29999  cvmlift2lem9a  30098  mbfresfi  32051  cocnv  32116  diophrw  35672  eldioph2  35675  mbfres2cn  37932  funcoressn  38773
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