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  7297  tposco  8258  mapen  9142  mapfien  9381  hashfacen  14521  relexpsucnnl  15105  relexpaddnn  15126  cofuass  17982  setccatid  18177  estrccatid  18224  frmdup3lem  18976  symggrplem  18994  f1omvdco2  19576  symggen  19598  psgnunilem1  19621  gsumval3  20035  gsumzf1o  20040  gsumzmhm  20065  prds1  20464  psrass1lem  22149  pf1mpf  22578  pf1ind  22581  qtophmeo  24044  uniioombllem2  25812  cncombf  25887  motgrp  28883  pjsdi2i  32624  pjadj2coi  32671  pj3lem1  32673  pj3i  32675  fcoinver  33064  fmptco1f1o  33093  fcobij  33178  fcobijfs  33179  cocnvf1o  33187  symgfcoeu  33509  pmtrcnel2  33517  cycpmconjv  33569  cycpmconjslem1  33581  cycpmconjs  33583  cyc3conja  33584  1arithidomlem2  33933  selvascl  34014  mplvrpmga  34042  mplvrpmrhm  34044  reprpmtf1o  35121  derangenlem  35737  subfacp1lem5  35750  erdsze2lem2  35770  pprodcnveq  36447  cocnv  38462  ltrncoidN  40988  trlcoabs2N  41582  trlcoat  41583  trlcone  41588  cdlemg46  41595  cdlemg47  41596  ltrnco4  41599  tgrpgrplem  41609  tendoplass  41643  cdlemi2  41679  cdlemk2  41692  cdlemk4  41694  cdlemk8  41698  cdlemk45  41807  cdlemk54  41818  cdlemk55a  41819  erngdvlem3  41850  erngdvlem3-rN  41858  tendocnv  41881  dvhvaddass  41957  dvhlveclem  41968  cdlemn8  42064  dihopelvalcpre  42108  dih1dimatlem0  42188  aks6d1c6lem5  43030  diophrw  43591  eldioph2  43594  mendring  44016  cortrcltrcl  44567  corclrtrcl  44568  cortrclrcl  44570  cotrclrtrcl  44571  cortrclrtrcl  44572  frege131d  44591  brcofffn  44858  brco3f1o  44860  neicvgnvo  44942  volicoff  46810  voliooicof  46811  ovolval4lem2  47465  3f1oss1  47950  gricushgr  48820  rngccatidALTV  49174  ringccatidALTV  49208  fuco11idx  50248