Theorem coass 6158

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 ∘ 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		relco 6137	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∘ 𝐶)
2		relco 6137	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ∘ 𝐶))
3		excom 2164	. . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
4		anass 468	. . . . 5 ⊢ (((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
5	4	2exbii 1852	. . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
6	3, 5	bitr4i 277	. . 3 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
7		vex 3426	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3426	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brco 5768	. . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))
10	9	anbi2i 622	. . . . 5 ⊢ ((𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
11	10	exbii 1851	. . . 4 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
12		vex 3426	. . . . 5 ⊢ 𝑥 ∈ V
13	12, 8	opelco 5769	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦))
14		exdistr 1959	. . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
15	11, 13, 14	3bitr4i 302	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
16		vex 3426	. . . . . . 7 ⊢ 𝑤 ∈ V
17	12, 16	brco 5768	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤))
18	17	anbi1i 623	. . . . 5 ⊢ ((𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
19	18	exbii 1851	. . . 4 ⊢ (∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
20	12, 8	opelco 5769	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦))
21		19.41v 1954	. . . . 5 ⊢ (∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
22	21	exbii 1851	. . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
23	19, 20, 22	3bitr4i 302	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
24	6, 15, 23	3bitr4i 302	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)))
25	1, 2, 24	eqrelriiv 5689	1 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070   ∘ ccom 5584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-co 5589

This theorem is referenced by:  funcoeqres  6730  fcof1oinvd  7145  tposco  8044  mapen  8877  mapfien  9097  hashfacen  14094  hashfacenOLD  14095  relexpsucnnl  14669  relexpaddnn  14690  cofuass  17520  setccatid  17715  estrccatid  17764  frmdup3lem  18420  symggrplem  18438  f1omvdco2  18971  symggen  18993  psgnunilem1  19016  gsumval3  19423  gsumzf1o  19428  gsumzmhm  19453  prds1  19768  psrass1lemOLD  21053  psrass1lem  21056  pf1mpf  21428  pf1ind  21431  qtophmeo  22876  uniioombllem2  24652  cncombf  24727  motgrp  26808  pjsdi2i  30420  pjadj2coi  30467  pj3lem1  30469  pj3i  30471  fcoinver  30847  fmptco1f1o  30869  fcobij  30959  fcobijfs  30960  symgfcoeu  31253  pmtrcnel2  31261  cycpmconjv  31311  cycpmconjslem1  31323  cycpmconjs  31325  cyc3conja  31326  reprpmtf1o  32506  derangenlem  33033  subfacp1lem5  33046  erdsze2lem2  33066  pprodcnveq  34112  cocnv  35810  ltrncoidN  38069  trlcoabs2N  38663  trlcoat  38664  trlcone  38669  cdlemg46  38676  cdlemg47  38677  ltrnco4  38680  tgrpgrplem  38690  tendoplass  38724  cdlemi2  38760  cdlemk2  38773  cdlemk4  38775  cdlemk8  38779  cdlemk45  38888  cdlemk54  38899  cdlemk55a  38900  erngdvlem3  38931  erngdvlem3-rN  38939  tendocnv  38962  dvhvaddass  39038  dvhlveclem  39049  cdlemn8  39145  dihopelvalcpre  39189  dih1dimatlem0  39269  diophrw  40497  eldioph2  40500  mendring  40933  cortrcltrcl  41237  corclrtrcl  41238  cortrclrcl  41240  cotrclrtrcl  41241  cortrclrtrcl  41242  frege131d  41261  brcofffn  41530  brco3f1o  41532  neicvgnvo  41614  volicoff  43426  voliooicof  43427  ovolval4lem2  44078  isomushgr  45166  rngccatidALTV  45435  ringccatidALTV  45498