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Theorem resco 5256
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 5062 . 2  |-  Rel  (
( A  o.  B
)  |`  C )
2 relco 5250 . 2  |-  Rel  ( A  o.  ( B  |`  C ) )
3 vex 2867 . . . . . 6  |-  x  e. 
_V
4 vex 2867 . . . . . 6  |-  y  e. 
_V
53, 4brco 4931 . . . . 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 1906 . . . 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 2867 . . . . . . . 8  |-  z  e. 
_V
109brres 5040 . . . . . . 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 1582 . . . 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 5040 . . 3  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  ( x ( A  o.  B ) y  /\  x  e.  C ) )
163, 4brco 4931 . . 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 4861 1  |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   class class class wbr 4102    |` cres 4770    o. ccom 4772
This theorem is referenced by:  cocnvcnv2  5263  coires1  5269  relcoi1  5280  dftpos2  6335  canthp1lem2  8362  o1res  12124  gsumzaddlem  15296  tsmsf1o  17923  tsmsmhm  17924  mbfres  19097  hhssims  21960  erdsze2lem2  24139  cvmlift2lem9a  24238  cocnv  25717  diophrw  26161  eldioph2  26164  funcoressn  27315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-co 4777  df-res 4780
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