Theorem coass 5220

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 5200	. 2 |- Rel ( ( A o. B ) o. C )
2		relco 5200	. 2 |- Rel ( A o. ( B o. C ) )
3		excom 1688	. . . 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 1630	. . . 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 2779	. . . . . . 7 |- z e. _V
8		vex 2779	. . . . . . 7 |- y e. _V
9	7, 8	brco 4867	. . . . . 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 1629	. . . 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 2779	. . . . 5 |- x e. _V
13	12, 8	opelco 4868	. . . 4 |- ( <. x , y >. e. ( ( A o. B ) o. C ) <-> E. z ( x C z /\ z ( A o. B ) y ) )
14		exdistr 1934	. . . 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 2779	. . . . . . 7 |- w e. _V
17	12, 16	brco 4867	. . . . . 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 1629	. . . 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 4868	. . . 4 |- ( <. x , y >. e. ( A o. ( B o. C ) ) <-> E. w ( x ( B o. C ) w /\ w A y ) )
21		19.41v 1927	. . . . 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 1629	. . . 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 4787	1 |- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) )

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2178   <.cop 3646   class class class wbr 4059   o. ccom 4697

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-co 4702

This theorem is referenced by:  funcoeqres  5575  fcof1o  5881  tposco  6384  mapen  6968  hashfacen  11018