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Theorem coass 4869
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 4849 . 2  |-  Rel  (
( A  o.  B
)  o.  C )
2 relco 4849 . 2  |-  Rel  ( A  o.  ( B  o.  C ) )
3 excom 1595 . . . 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 393 . . . . 5  |-  ( ( ( x C z  /\  z B w )  /\  w A y )  <->  ( x C z  /\  (
z B w  /\  w A y ) ) )
542exbii 1538 . . . 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 185 . . 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 2605 . . . . . . 7  |-  z  e. 
_V
8 vex 2605 . . . . . . 7  |-  y  e. 
_V
97, 8brco 4534 . . . . . 6  |-  ( z ( A  o.  B
) y  <->  E. w
( z B w  /\  w A y ) )
109anbi2i 445 . . . . 5  |-  ( ( x C z  /\  z ( A  o.  B ) y )  <-> 
( x C z  /\  E. w ( z B w  /\  w A y ) ) )
1110exbii 1537 . . . 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 2605 . . . . 5  |-  x  e. 
_V
1312, 8opelco 4535 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z ( x C z  /\  z
( A  o.  B
) y ) )
14 exdistr 1829 . . . 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 210 . . 3  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z E. w
( x C z  /\  ( z B w  /\  w A y ) ) )
16 vex 2605 . . . . . . 7  |-  w  e. 
_V
1712, 16brco 4534 . . . . . 6  |-  ( x ( B  o.  C
) w  <->  E. z
( x C z  /\  z B w ) )
1817anbi1i 446 . . . . 5  |-  ( ( x ( B  o.  C ) w  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
1918exbii 1537 . . . 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 4535 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  ( B  o.  C )
)  <->  E. w ( x ( B  o.  C
) w  /\  w A y ) )
21 19.41v 1824 . . . . 5  |-  ( E. z ( ( x C z  /\  z B w )  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
2221exbii 1537 . . . 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 210 . . 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 210 . 2  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  <. x ,  y
>.  e.  ( A  o.  ( B  o.  C
) ) )
251, 2, 24eqrelriiv 4460 1  |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   <.cop 3409   class class class wbr 3793    o. ccom 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-co 4380
This theorem is referenced by:  funcoeqres  5188  fcof1o  5460  tposco  5924
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