Theorem coass 5255

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 5235	. 2 |- Rel ( ( A o. B ) o. C )
2		relco 5235	. 2 |- Rel ( A o. ( B o. C ) )
3		excom 1712	. . . 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 401	. . . . 5 |- ( ( ( x C z /\ z B w ) /\ w A y ) <-> ( x C z /\ ( z B w /\ w A y ) ) )
5	4	2exbii 1654	. . . 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 187	. . 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 2805	. . . . . . 7 |- z e. _V
8		vex 2805	. . . . . . 7 |- y e. _V
9	7, 8	brco 4901	. . . . . 6 |- ( z ( A o. B ) y <-> E. w ( z B w /\ w A y ) )
10	9	anbi2i 457	. . . . 5 |- ( ( x C z /\ z ( A o. B ) y ) <-> ( x C z /\ E. w ( z B w /\ w A y ) ) )
11	10	exbii 1653	. . . 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 2805	. . . . 5 |- x e. _V
13	12, 8	opelco 4902	. . . 4 |- ( <. x , y >. e. ( ( A o. B ) o. C ) <-> E. z ( x C z /\ z ( A o. B ) y ) )
14		exdistr 1958	. . . 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 212	. . 3 |- ( <. x , y >. e. ( ( A o. B ) o. C ) <-> E. z E. w ( x C z /\ ( z B w /\ w A y ) ) )
16		vex 2805	. . . . . . 7 |- w e. _V
17	12, 16	brco 4901	. . . . . 6 |- ( x ( B o. C ) w <-> E. z ( x C z /\ z B w ) )
18	17	anbi1i 458	. . . . 5 |- ( ( x ( B o. C ) w /\ w A y ) <-> ( E. z ( x C z /\ z B w ) /\ w A y ) )
19	18	exbii 1653	. . . 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 4902	. . . 4 |- ( <. x , y >. e. ( A o. ( B o. C ) ) <-> E. w ( x ( B o. C ) w /\ w A y ) )
21		19.41v 1951	. . . . 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 1653	. . . 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 212	. . 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 212	. 2 |- ( <. x , y >. e. ( ( A o. B ) o. C ) <-> <. x , y >. e. ( A o. ( B o. C ) ) )
25	1, 2, 24	eqrelriiv 4820	1 |- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) )

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   <.cop 3672   class class class wbr 4088   o. ccom 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-co 4734

This theorem is referenced by:  funcoeqres  5614  fcof1o  5930  tposco  6441  mapen  7032  hashfacen  11100