Theorem coass 6223

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 6066	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∘ 𝐶)
2		relco 6066	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ∘ 𝐶))
3		excom 2168	. . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
4		anass 468	. . . . 5 ⊢ (((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
5	4	2exbii 1851	. . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
6	3, 5	bitr4i 278	. . 3 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
7		vex 3443	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3443	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brco 5818	. . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))
10	9	anbi2i 624	. . . . 5 ⊢ ((𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
11	10	exbii 1850	. . . 4 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
12		vex 3443	. . . . 5 ⊢ 𝑥 ∈ V
13	12, 8	opelco 5819	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦))
14		exdistr 1956	. . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
15	11, 13, 14	3bitr4i 303	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)))
16		vex 3443	. . . . . . 7 ⊢ 𝑤 ∈ V
17	12, 16	brco 5818	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤))
18	17	anbi1i 625	. . . . 5 ⊢ ((𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
19	18	exbii 1850	. . . 4 ⊢ (∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
20	12, 8	opelco 5819	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦))
21		19.41v 1951	. . . . 5 ⊢ (∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
22	21	exbii 1850	. . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
23	19, 20, 22	3bitr4i 303	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
24	6, 15, 23	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)))
25	1, 2, 24	eqrelriiv 5738	1 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ⟨cop 4585   class class class wbr 5097   ∘ ccom 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-co 5632

This theorem is referenced by:  funcoeqres  6804  fcof1oinvd  7239  tposco  8199  mapen  9071  mapfien  9313  hashfacen  14379  relexpsucnnl  14955  relexpaddnn  14976  cofuass  17815  setccatid  18010  estrccatid  18057  frmdup3lem  18793  symggrplem  18811  f1omvdco2  19379  symggen  19401  psgnunilem1  19424  gsumval3  19838  gsumzf1o  19843  gsumzmhm  19868  prds1  20260  psrass1lem  21890  pf1mpf  22298  pf1ind  22301  qtophmeo  23763  uniioombllem2  25542  cncombf  25617  motgrp  28596  pjsdi2i  32213  pjadj2coi  32260  pj3lem1  32262  pj3i  32264  fcoinver  32659  fmptco1f1o  32691  fcobij  32778  fcobijfs  32779  cocnvf1o  32787  symgfcoeu  33143  pmtrcnel2  33151  cycpmconjv  33203  cycpmconjslem1  33215  cycpmconjs  33217  cyc3conja  33218  1arithidomlem2  33596  mplvrpmga  33689  mplvrpmrhm  33691  reprpmtf1o  34762  derangenlem  35344  subfacp1lem5  35357  erdsze2lem2  35377  pprodcnveq  36054  cocnv  37895  ltrncoidN  40423  trlcoabs2N  41017  trlcoat  41018  trlcone  41023  cdlemg46  41030  cdlemg47  41031  ltrnco4  41034  tgrpgrplem  41044  tendoplass  41078  cdlemi2  41114  cdlemk2  41127  cdlemk4  41129  cdlemk8  41133  cdlemk45  41242  cdlemk54  41253  cdlemk55a  41254  erngdvlem3  41285  erngdvlem3-rN  41293  tendocnv  41316  dvhvaddass  41392  dvhlveclem  41403  cdlemn8  41499  dihopelvalcpre  41543  dih1dimatlem0  41623  aks6d1c6lem5  42466  diophrw  43038  eldioph2  43041  mendring  43467  cortrcltrcl  44018  corclrtrcl  44019  cortrclrcl  44021  cotrclrtrcl  44022  cortrclrtrcl  44023  frege131d  44042  brcofffn  44309  brco3f1o  44311  neicvgnvo  44393  volicoff  46276  voliooicof  46277  ovolval4lem2  46931  3f1oss1  47358  gricushgr  48200  rngccatidALTV  48555  ringccatidALTV  48589  fuco11idx  49617