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Theorem coass 5189
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 )
)
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.

Proof of Theorem coass
StepHypRef Expression
1 relco 5169 . 2  |-  Rel  (
( A  o.  B
)  o.  C )
2 relco 5169 . 2  |-  Rel  ( A  o.  ( B  o.  C ) )
3 excom 1787 . . . 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 632 . . . . 5  |-  ( ( ( x C z  /\  z B w )  /\  w A y )  <->  ( x C z  /\  (
z B w  /\  w A y ) ) )
542exbii 1571 . . . 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 245 . . 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 2792 . . . . . . 7  |-  z  e. 
_V
8 vex 2792 . . . . . . 7  |-  y  e. 
_V
97, 8brco 4851 . . . . . 6  |-  ( z ( A  o.  B
) y  <->  E. w
( z B w  /\  w A y ) )
109anbi2i 677 . . . . 5  |-  ( ( x C z  /\  z ( A  o.  B ) y )  <-> 
( x C z  /\  E. w ( z B w  /\  w A y ) ) )
1110exbii 1570 . . . 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 2792 . . . . 5  |-  x  e. 
_V
1312, 8opelco 4852 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z ( x C z  /\  z
( A  o.  B
) y ) )
14 exdistr 1848 . . . 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 270 . . 3  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z E. w
( x C z  /\  ( z B w  /\  w A y ) ) )
16 vex 2792 . . . . . . 7  |-  w  e. 
_V
1712, 16brco 4851 . . . . . 6  |-  ( x ( B  o.  C
) w  <->  E. z
( x C z  /\  z B w ) )
1817anbi1i 678 . . . . 5  |-  ( ( x ( B  o.  C ) w  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
1918exbii 1570 . . . 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 4852 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  ( B  o.  C )
)  <->  E. w ( x ( B  o.  C
) w  /\  w A y ) )
21 19.41v 1843 . . . . 5  |-  ( E. z ( ( x C z  /\  z B w )  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
2221exbii 1570 . . . 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 270 . . 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 270 . 2  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  <. x ,  y
>.  e.  ( A  o.  ( B  o.  C
) ) )
251, 2, 24eqrelriiv 4780 1  |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   <.cop 3644   class class class wbr 4024    o. ccom 4692
This theorem is referenced by:  funcoeqres  5469  fcof1o  5764  tposco  6226  mapen  7020  mapfien  7394  hashfacen  11386  cofuass  13757  setccatid  13910  frmdup3  14482  symggrp  14774  gsumval3  15185  gsumzf1o  15190  gsumzmhm  15204  prds1  15391  psrass1lem  16117  qtophmeo  17502  uniioombllem2  18932  cncombf  19007  pf1mpf  19429  pf1ind  19432  pjsdi2i  22729  pjadj2coi  22776  pj3lem1  22778  pj3i  22780  derangenlem  23106  subfacp1lem5  23119  erdsze2lem2  23139  relexpsucl  23432  relexpadd  23439  pprodcnveq  23831  hmeogrpi  24935  cmpmorass  25365  cocnv  25792  diophrw  26237  eldioph2  26240  f1omvdco2  26790  symggen  26810  psgnunilem1  26815  mendrng  26899  ltrncoidN  29584  trlcoabs2N  30178  trlcoat  30179  trlcone  30184  cdlemg46  30191  cdlemg47  30192  ltrnco4  30195  tgrpgrplem  30205  tendoplass  30239  cdlemi2  30275  cdlemk2  30288  cdlemk4  30290  cdlemk8  30294  cdlemk45  30403  cdlemk54  30414  cdlemk55a  30415  erngdvlem3  30446  erngdvlem3-rN  30454  tendocnv  30478  dvhvaddass  30554  dvhlveclem  30565  cdlemn8  30661  dihopelvalcpre  30705  dih1dimatlem0  30785
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-co 4697
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