Theorem coass 5097

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 set.mm contributors, 27-Jan-1997.)

Assertion

Ref	Expression
coass	⊢ ((A ∘ B) ∘ C) = (A ∘ (B ∘ 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		excom 1741	. . . . 5 ⊢ (∃w∃z((xCz ∧ zBw) ∧ wAy) ↔ ∃z∃w((xCz ∧ zBw) ∧ wAy))
2		anass 630	. . . . . 6 ⊢ (((xCz ∧ zBw) ∧ wAy) ↔ (xCz ∧ (zBw ∧ wAy)))
3	2	2exbii 1583	. . . . 5 ⊢ (∃z∃w((xCz ∧ zBw) ∧ wAy) ↔ ∃z∃w(xCz ∧ (zBw ∧ wAy)))
4	1, 3	bitr2i 241	. . . 4 ⊢ (∃z∃w(xCz ∧ (zBw ∧ wAy)) ↔ ∃w∃z((xCz ∧ zBw) ∧ wAy))
5		brco 4883	. . . . . . 7 ⊢ (z(A ∘ B)y ↔ ∃w(zBw ∧ wAy))
6	5	anbi2i 675	. . . . . 6 ⊢ ((xCz ∧ z(A ∘ B)y) ↔ (xCz ∧ ∃w(zBw ∧ wAy)))
7		19.42v 1905	. . . . . 6 ⊢ (∃w(xCz ∧ (zBw ∧ wAy)) ↔ (xCz ∧ ∃w(zBw ∧ wAy)))
8	6, 7	bitr4i 243	. . . . 5 ⊢ ((xCz ∧ z(A ∘ B)y) ↔ ∃w(xCz ∧ (zBw ∧ wAy)))
9	8	exbii 1582	. . . 4 ⊢ (∃z(xCz ∧ z(A ∘ B)y) ↔ ∃z∃w(xCz ∧ (zBw ∧ wAy)))
10		brco 4883	. . . . . . 7 ⊢ (x(B ∘ C)w ↔ ∃z(xCz ∧ zBw))
11	10	anbi1i 676	. . . . . 6 ⊢ ((x(B ∘ C)w ∧ wAy) ↔ (∃z(xCz ∧ zBw) ∧ wAy))
12		19.41v 1901	. . . . . 6 ⊢ (∃z((xCz ∧ zBw) ∧ wAy) ↔ (∃z(xCz ∧ zBw) ∧ wAy))
13	11, 12	bitr4i 243	. . . . 5 ⊢ ((x(B ∘ C)w ∧ wAy) ↔ ∃z((xCz ∧ zBw) ∧ wAy))
14	13	exbii 1582	. . . 4 ⊢ (∃w(x(B ∘ C)w ∧ wAy) ↔ ∃w∃z((xCz ∧ zBw) ∧ wAy))
15	4, 9, 14	3bitr4i 268	. . 3 ⊢ (∃z(xCz ∧ z(A ∘ B)y) ↔ ∃w(x(B ∘ C)w ∧ wAy))
16		brco 4883	. . 3 ⊢ (x((A ∘ B) ∘ C)y ↔ ∃z(xCz ∧ z(A ∘ B)y))
17		brco 4883	. . 3 ⊢ (x(A ∘ (B ∘ C))y ↔ ∃w(x(B ∘ C)w ∧ wAy))
18	15, 16, 17	3bitr4i 268	. 2 ⊢ (x((A ∘ B) ∘ C)y ↔ x(A ∘ (B ∘ C))y)
19	18	eqbrriv 4851	1 ⊢ ((A ∘ B) ∘ C) = (A ∘ (B ∘ C))

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   class class class wbr 4639   ∘ ccom 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726

This theorem is referenced by:  cnvpprod  5841  enmap2lem3  6065  enmap2lem5  6067  enmap1lem3  6071  enmap1lem5  6073