Theorem coass 6266

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 6108	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∘ 𝐶)
2		relco 6108	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ∘ 𝐶))
3		excom 2199	. . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
4		anass 474	. . . . 5 ⊢ (((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
5	4	2exbii 1882	. . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
6	3, 5	bitr4i 281	. . 3 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
7		vex 3457	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3457	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brco 5854	. . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))
10	9	anbi2i 635	. . . . 5 ⊢ ((𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
11	10	exbii 1881	. . . 4 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
12		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
13	12, 8	opelco 5855	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦))
14		exdistr 1987	. . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
15	11, 13, 14	3bitr4i 306	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
16		vex 3457	. . . . . . 7 ⊢ 𝑤 ∈ V
17	12, 16	brco 5854	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤))
18	17	anbi1i 636	. . . . 5 ⊢ ((𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
19	18	exbii 1881	. . . 4 ⊢ (∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
20	12, 8	opelco 5855	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦))
21		19.41v 1982	. . . . 5 ⊢ (∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
22	21	exbii 1881	. . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
23	19, 20, 22	3bitr4i 306	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
24	6, 15, 23	3bitr4i 306	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)))
25	1, 2, 24	eqrelriiv 5774	1 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ⟨cop 4593   class class class wbr 5107   ∘ ccom 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-co 5668

This theorem is used by:  funcoeqres  6853  fcof1oinvd  7298  tposco  8259  mapen  9143  mapfien  9382  hashfacen  14523  relexpsucnnl  15107  relexpaddnn  15128  cofuass  17984  setccatid  18179  estrccatid  18226  frmdup3lem  18981  symggrplem  18999  f1omvdco2  19581  symggen  19603  psgnunilem1  19626  gsumval3  20040  gsumzf1o  20045  gsumzmhm  20070  prds1  20469  psrass1lem  22154  pf1mpf  22583  pf1ind  22586  qtophmeo  24049  uniioombllem2  25817  cncombf  25892  motgrp  28893  pjsdi2i  32646  pjadj2coi  32693  pj3lem1  32695  pj3i  32697  fcoinver  33085  fmptco1f1o  33114  fcobij  33199  fcobijfs  33200  cocnvf1o  33208  symgfcoeu  33530  pmtrcnel2  33538  cycpmconjv  33590  cycpmconjslem1  33602  cycpmconjs  33604  cyc3conja  33605  1arithidomlem2  33954  selvascl  34035  mplvrpmga  34063  mplvrpmrhm  34065  reprpmtf1o  35142  derangenlem  35758  subfacp1lem5  35771  erdsze2lem2  35791  pprodcnveq  36468  cocnv  38483  ltrncoidN  41009  trlcoabs2N  41603  trlcoat  41604  trlcone  41609  cdlemg46  41616  cdlemg47  41617  ltrnco4  41620  tgrpgrplem  41630  tendoplass  41664  cdlemi2  41700  cdlemk2  41713  cdlemk4  41715  cdlemk8  41719  cdlemk45  41828  cdlemk54  41839  cdlemk55a  41840  erngdvlem3  41871  erngdvlem3-rN  41879  tendocnv  41902  dvhvaddass  41978  dvhlveclem  41989  cdlemn8  42085  dihopelvalcpre  42129  dih1dimatlem0  42209  aks6d1c6lem5  43051  diophrw  43612  eldioph2  43615  mendring  44037  cortrcltrcl  44588  corclrtrcl  44589  cortrclrcl  44591  cotrclrtrcl  44592  cortrclrtrcl  44593  frege131d  44612  brcofffn  44879  brco3f1o  44881  neicvgnvo  44963  volicoff  46831  voliooicof  46832  ovolval4lem2  47486  3f1oss1  47971  gricushgr  48841  rngccatidALTV  49195  ringccatidALTV  49229  fuco11idx  50269