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Theorem coass 5129
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 5109 . 2  |-  Rel  (
( A  o.  B
)  o.  C )
2 relco 5109 . 2  |-  Rel  ( A  o.  ( B  o.  C ) )
3 excom 1657 . . . 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 399 . . . . 5  |-  ( ( ( x C z  /\  z B w )  /\  w A y )  <->  ( x C z  /\  (
z B w  /\  w A y ) ) )
542exbii 1599 . . . 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 186 . . 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 2733 . . . . . . 7  |-  z  e. 
_V
8 vex 2733 . . . . . . 7  |-  y  e. 
_V
97, 8brco 4782 . . . . . 6  |-  ( z ( A  o.  B
) y  <->  E. w
( z B w  /\  w A y ) )
109anbi2i 454 . . . . 5  |-  ( ( x C z  /\  z ( A  o.  B ) y )  <-> 
( x C z  /\  E. w ( z B w  /\  w A y ) ) )
1110exbii 1598 . . . 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 2733 . . . . 5  |-  x  e. 
_V
1312, 8opelco 4783 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z ( x C z  /\  z
( A  o.  B
) y ) )
14 exdistr 1902 . . . 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 211 . . 3  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  E. z E. w
( x C z  /\  ( z B w  /\  w A y ) ) )
16 vex 2733 . . . . . . 7  |-  w  e. 
_V
1712, 16brco 4782 . . . . . 6  |-  ( x ( B  o.  C
) w  <->  E. z
( x C z  /\  z B w ) )
1817anbi1i 455 . . . . 5  |-  ( ( x ( B  o.  C ) w  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
1918exbii 1598 . . . 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 4783 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  ( B  o.  C )
)  <->  E. w ( x ( B  o.  C
) w  /\  w A y ) )
21 19.41v 1895 . . . . 5  |-  ( E. z ( ( x C z  /\  z B w )  /\  w A y )  <->  ( E. z ( x C z  /\  z B w )  /\  w A y ) )
2221exbii 1598 . . . 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 211 . . 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 211 . 2  |-  ( <.
x ,  y >.  e.  ( ( A  o.  B )  o.  C
)  <->  <. x ,  y
>.  e.  ( A  o.  ( B  o.  C
) ) )
251, 2, 24eqrelriiv 4705 1  |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   E.wex 1485    e. wcel 2141   <.cop 3586   class class class wbr 3989    o. ccom 4615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-co 4620
This theorem is referenced by:  funcoeqres  5473  fcof1o  5768  tposco  6254  mapen  6824  hashfacen  10771
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