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Theorem coass 5207
Description: Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
Assertion
Ref Expression
coass  |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C )
)

Proof of Theorem coass
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5187 . 2  |-  Rel  (
( A  o.  B
)  o.  C )
2 relco 5187 . 2  |-  Rel  ( A  o.  ( B  o.  C ) )
3 excom 1798 . . . 4  |-  ( E. z E. w ( x C z  /\  ( z B w  /\  w A y ) )  <->  E. w E. z ( x C z  /\  ( z B w  /\  w A y ) ) )
4 anass 630 . . . . 5  |-  ( ( ( x C z  /\  z B w )  /\  w A y )  <->  ( x C z  /\  (
z B w  /\  w A y ) ) )
542exbii 1573 . . . 4  |-  ( E. w E. z ( ( x C z  /\  z B w )  /\  w A y )  <->  E. w E. z ( x C z  /\  ( z B w  /\  w A y ) ) )
63, 5bitr4i 243 . . 3  |-  ( E. z E. w ( x C z  /\  ( z B w  /\  w A y ) )  <->  E. w E. z ( ( x C z  /\  z B w )  /\  w A y ) )
7 vex 2804 . . . . . . 7  |-  z  e. 
_V
8 vex 2804 . . . . . . 7  |-  y  e. 
_V
97, 8brco 4868 . . . . . 6  |-  ( z ( A  o.  B
) y  <->  E. w
( z B w  /\  w A y ) )
109anbi2i 675 . . . . 5  |-  ( ( x C z  /\  z ( A  o.  B ) y )  <-> 
( x C z  /\  E. w ( z B w  /\  w A y ) ) )
1110exbii 1572 . . . 4  |-  ( E. z ( x C z  /\  z ( A  o.  B ) y )  <->  E. z
( x C z  /\  E. w ( z B w  /\  w A y ) ) )
12 vex 2804 . . . . 5  |-  x  e. 
_V
1312, 8opelco 4869 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z ( x C z  /\  z
( A  o.  B
) y ) )
14 exdistr 1859 . . . 4  |-  ( E. z E. w ( x C z  /\  ( z B w  /\  w A y ) )  <->  E. z
( x C z  /\  E. w ( z B w  /\  w A y ) ) )
1511, 13, 143bitr4i 268 . . 3  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z E. w
( x C z  /\  ( z B w  /\  w A y ) ) )
16 vex 2804 . . . . . . 7  |-  w  e. 
_V
1712, 16brco 4868 . . . . . 6  |-  ( x ( B  o.  C
) w  <->  E. z
( x C z  /\  z B w ) )
1817anbi1i 676 . . . . 5  |-  ( ( x ( B  o.  C ) w  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
1918exbii 1572 . . . 4  |-  ( E. w ( x ( B  o.  C ) w  /\  w A y )  <->  E. w
( E. z ( x C z  /\  z B w )  /\  w A y ) )
2012, 8opelco 4869 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  ( B  o.  C )
)  <->  E. w ( x ( B  o.  C
) w  /\  w A y ) )
21 19.41v 1854 . . . . 5  |-  ( E. z ( ( x C z  /\  z B w )  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
2221exbii 1572 . . . 4  |-  ( E. w E. z ( ( x C z  /\  z B w )  /\  w A y )  <->  E. w
( E. z ( x C z  /\  z B w )  /\  w A y ) )
2319, 20, 223bitr4i 268 . . 3  |-  ( <.
x ,  y >.  e.  ( A  o.  ( B  o.  C )
)  <->  E. w E. z
( ( x C z  /\  z B w )  /\  w A y ) )
246, 15, 233bitr4i 268 . 2  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  <. x ,  y
>.  e.  ( A  o.  ( B  o.  C
) ) )
251, 2, 24eqrelriiv 4797 1  |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039    o. ccom 4709
This theorem is referenced by:  funcoeqres  5520  fcof1o  5819  tposco  6281  mapen  7041  mapfien  7415  hashfacen  11408  cofuass  13779  setccatid  13932  frmdup3  14504  symggrp  14796  gsumval3  15207  gsumzf1o  15212  gsumzmhm  15226  prds1  15413  psrass1lem  16139  qtophmeo  17524  uniioombllem2  18954  cncombf  19029  pf1mpf  19451  pf1ind  19454  pjsdi2i  22753  pjadj2coi  22800  pj3lem1  22802  pj3i  22804  derangenlem  23717  subfacp1lem5  23730  erdsze2lem2  23750  relexpsucl  24043  relexpadd  24050  pprodcnveq  24494  hmeogrpi  25639  cmpmorass  26069  cocnv  26496  diophrw  26941  eldioph2  26944  f1omvdco2  27494  symggen  27514  psgnunilem1  27519  mendrng  27603  ltrncoidN  30939  trlcoabs2N  31533  trlcoat  31534  trlcone  31539  cdlemg46  31546  cdlemg47  31547  ltrnco4  31550  tgrpgrplem  31560  tendoplass  31594  cdlemi2  31630  cdlemk2  31643  cdlemk4  31645  cdlemk8  31649  cdlemk45  31758  cdlemk54  31769  cdlemk55a  31770  erngdvlem3  31801  erngdvlem3-rN  31809  tendocnv  31833  dvhvaddass  31909  dvhlveclem  31920  cdlemn8  32016  dihopelvalcpre  32060  dih1dimatlem0  32140
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-co 4714
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