Theorem coass 5122

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.

Step	Hyp	Ref	Expression
1		relco 5102	. 2 |- Rel ( ( A o. B ) o. C )
2		relco 5102	. 2 |- Rel ( A o. ( B o. C ) )
3		excom 1652	. . . 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 ) ) )
5	4	2exbii 1594	. . . 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 ) ) )
6	3, 5	bitr4i 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 2729	. . . . . . 7 |- z e. _V
8		vex 2729	. . . . . . 7 |- y e. _V
9	7, 8	brco 4775	. . . . . 6 |- ( z ( A o. B ) y <-> E. w ( z B w /\ w A y ) )
10	9	anbi2i 453	. . . . 5 |- ( ( x C z /\ z ( A o. B ) y ) <-> ( x C z /\ E. w ( z B w /\ w A y ) ) )
11	10	exbii 1593	. . . 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 2729	. . . . 5 |- x e. _V
13	12, 8	opelco 4776	. . . 4 |- ( <. x , y >. e. ( ( A o. B ) o. C ) <-> E. z ( x C z /\ z ( A o. B ) y ) )
14		exdistr 1897	. . . 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 ) ) )
15	11, 13, 14	3bitr4i 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 2729	. . . . . . 7 |- w e. _V
17	12, 16	brco 4775	. . . . . 6 |- ( x ( B o. C ) w <-> E. z ( x C z /\ z B w ) )
18	17	anbi1i 454	. . . . 5 |- ( ( x ( B o. C ) w /\ w A y ) <-> ( E. z ( x C z /\ z B w ) /\ w A y ) )
19	18	exbii 1593	. . . 4 |- ( E. w ( x ( B o. C ) w /\ w A y ) <-> E. w ( E. z ( x C z /\ z B w ) /\ w A y ) )
20	12, 8	opelco 4776	. . . 4 |- ( <. x , y >. e. ( A o. ( B o. C ) ) <-> E. w ( x ( B o. C ) w /\ w A y ) )
21		19.41v 1890	. . . . 5 |- ( E. z ( ( x C z /\ z B w ) /\ w A y ) <-> ( E. z ( x C z /\ z B w ) /\ w A y ) )
22	21	exbii 1593	. . . 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 ) )
23	19, 20, 22	3bitr4i 211	. . 3 |- ( <. x , y >. e. ( A o. ( B o. C ) ) <-> E. w E. z ( ( x C z /\ z B w ) /\ w A y ) )
24	6, 15, 23	3bitr4i 211	. 2 |- ( <. x , y >. e. ( ( A o. B ) o. C ) <-> <. x , y >. e. ( A o. ( B o. C ) ) )
25	1, 2, 24	eqrelriiv 4698	1 |- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) )

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   <.cop 3579   class class class wbr 3982   o. ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-co 4613

This theorem is referenced by:  funcoeqres  5463  fcof1o  5757  tposco  6243  mapen  6812  hashfacen  10749