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Theorem resco 5365
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 5165 . 2  |-  Rel  (
( A  o.  B
)  |`  C )
2 relco 5359 . 2  |-  Rel  ( A  o.  ( B  |`  C ) )
3 vex 2951 . . . . . 6  |-  x  e. 
_V
4 vex 2951 . . . . . 6  |-  y  e. 
_V
53, 4brco 5034 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
65anbi1i 677 . . . 4  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  ( E. z ( x B z  /\  z A y )  /\  x  e.  C )
)
7 19.41v 1924 . . . 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 774 . . . . . 6  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
9 vex 2951 . . . . . . . 8  |-  z  e. 
_V
109brres 5143 . . . . . . 7  |-  ( x ( B  |`  C ) z  <->  ( x B z  /\  x  e.  C ) )
1110anbi1i 677 . . . . . 6  |-  ( ( x ( B  |`  C ) z  /\  z A y )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
128, 11bitr4i 244 . . . . 5  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( x
( B  |`  C ) z  /\  z A y ) )
1312exbii 1592 . . . 4  |-  ( E. z ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
146, 7, 133bitr2i 265 . . 3  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
154brres 5143 . . 3  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  ( x ( A  o.  B ) y  /\  x  e.  C ) )
163, 4brco 5034 . . 3  |-  ( x ( A  o.  ( B  |`  C ) ) y  <->  E. z ( x ( B  |`  C ) z  /\  z A y ) )
1714, 15, 163bitr4i 269 . 2  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  x ( A  o.  ( B  |`  C ) ) y )
181, 2, 17eqbrriv 4962 1  |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   class class class wbr 4204    |` cres 4871    o. ccom 4873
This theorem is referenced by:  cocnvcnv2  5372  coires1  5378  relcoi1  5389  dftpos2  6487  canthp1lem2  8517  o1res  12342  gsumzaddlem  15514  tsmsf1o  18162  tsmsmhm  18163  mbfres  19524  hhssims  22763  erdsze2lem2  24878  cvmlift2lem9a  24978  mbfresfi  26199  cocnv  26364  diophrw  26754  eldioph2  26757  funcoressn  27905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-co 4878  df-res 4881
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