Theorem coass 6268

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 6111	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∘ 𝐶)
2		relco 6111	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ∘ 𝐶))
3		excom 2203	. . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
4		anass 473	. . . . 5 ⊢ (((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
5	4	2exbii 1876	. . . 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 3465	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3465	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brco 5857	. . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))
10	9	anbi2i 634	. . . . 5 ⊢ ((𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
11	10	exbii 1875	. . . 4 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
12		vex 3465	. . . . 5 ⊢ 𝑥 ∈ V
13	12, 8	opelco 5858	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦))
14		exdistr 1981	. . . 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 3465	. . . . . . 7 ⊢ 𝑤 ∈ V
17	12, 16	brco 5857	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤))
18	17	anbi1i 635	. . . . 5 ⊢ ((𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
19	18	exbii 1875	. . . 4 ⊢ (∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
20	12, 8	opelco 5858	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦))
21		19.41v 1976	. . . . 5 ⊢ (∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
22	21	exbii 1875	. . . 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 5777	1 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ⟨cop 4598   class class class wbr 5111   ∘ ccom 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5259  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-xp 5668  df-rel 5669  df-co 5671

This theorem is referenced by:  funcoeqres  6853  fcof1oinvd  7292  tposco  8253  mapen  9129  mapfien  9368  hashfacen  14491  relexpsucnnl  15067  relexpaddnn  15088  cofuass  17946  setccatid  18141  estrccatid  18188  frmdup3lem  18925  symggrplem  18943  f1omvdco2  19518  symggen  19540  psgnunilem1  19563  gsumval3  19977  gsumzf1o  19982  gsumzmhm  20007  prds1  20404  psrass1lem  22052  pf1mpf  22481  pf1ind  22484  qtophmeo  23943  uniioombllem2  25711  cncombf  25786  motgrp  28778  pjsdi2i  32450  pjadj2coi  32497  pj3lem1  32499  pj3i  32501  fcoinver  32890  fmptco1f1o  32919  fcobij  33006  fcobijfs  33007  cocnvf1o  33015  symgfcoeu  33343  pmtrcnel2  33351  cycpmconjv  33403  cycpmconjslem1  33415  cycpmconjs  33417  cyc3conja  33418  1arithidomlem2  33771  selvascl  33852  mplvrpmga  33880  mplvrpmrhm  33882  reprpmtf1o  34958  derangenlem  35596  subfacp1lem5  35609  erdsze2lem2  35629  pprodcnveq  36306  cocnv  38299  ltrncoidN  40827  trlcoabs2N  41421  trlcoat  41422  trlcone  41427  cdlemg46  41434  cdlemg47  41435  ltrnco4  41438  tgrpgrplem  41448  tendoplass  41482  cdlemi2  41518  cdlemk2  41531  cdlemk4  41533  cdlemk8  41537  cdlemk45  41646  cdlemk54  41657  cdlemk55a  41658  erngdvlem3  41689  erngdvlem3-rN  41697  tendocnv  41720  dvhvaddass  41796  dvhlveclem  41807  cdlemn8  41903  dihopelvalcpre  41947  dih1dimatlem0  42027  aks6d1c6lem5  42869  diophrw  43417  eldioph2  43420  mendring  43842  cortrcltrcl  44393  corclrtrcl  44394  cortrclrcl  44396  cotrclrtrcl  44397  cortrclrtrcl  44398  frege131d  44417  brcofffn  44684  brco3f1o  44686  neicvgnvo  44768  volicoff  46636  voliooicof  46637  ovolval4lem2  47291  3f1oss1  47736  gricushgr  48606  rngccatidALTV  48961  ringccatidALTV  48995  fuco11idx  50033