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Mirrors > Home > MPE Home > Th. List > coass | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
coass | ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 6107 | . 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∘ 𝐶) | |
2 | relco 6107 | . 2 ⊢ Rel (𝐴 ∘ (𝐵 ∘ 𝐶)) | |
3 | excom 2161 | . . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) | |
4 | anass 468 | . . . . 5 ⊢ (((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) | |
5 | 4 | 2exbii 1850 | . . . 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 3477 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
8 | vex 3477 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brco 5870 | . . . . . 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 1849 | . . . 4 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) |
12 | vex 3477 | . . . . 5 ⊢ 𝑥 ∈ V | |
13 | 12, 8 | opelco 5871 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦)) |
14 | exdistr 1957 | . . . 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 3477 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
17 | 12, 16 | brco 5870 | . . . . . 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 1849 | . . . 4 ⊢ (∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) |
20 | 12, 8 | opelco 5871 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦)) |
21 | 19.41v 1952 | . . . . 5 ⊢ (∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) | |
22 | 21 | exbii 1849 | . . . 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 5790 | 1 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1780 ∈ wcel 2105 〈cop 4634 class class class wbr 5148 ∘ ccom 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-11 2153 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-co 5685 |
This theorem is referenced by: funcoeqres 6864 fcof1oinvd 7294 tposco 8248 mapen 9147 mapfien 9409 hashfacen 14420 hashfacenOLD 14421 relexpsucnnl 14984 relexpaddnn 15005 cofuass 17846 setccatid 18044 estrccatid 18093 frmdup3lem 18789 symggrplem 18807 f1omvdco2 19364 symggen 19386 psgnunilem1 19409 gsumval3 19823 gsumzf1o 19828 gsumzmhm 19853 prds1 20218 psrass1lemOLD 21803 psrass1lem 21806 pf1mpf 22191 pf1ind 22194 qtophmeo 23641 uniioombllem2 25432 cncombf 25507 motgrp 28228 pjsdi2i 31844 pjadj2coi 31891 pj3lem1 31893 pj3i 31895 fcoinver 32269 fmptco1f1o 32291 fcobij 32381 fcobijfs 32382 symgfcoeu 32680 pmtrcnel2 32688 cycpmconjv 32738 cycpmconjslem1 32750 cycpmconjs 32752 cyc3conja 32753 reprpmtf1o 34103 derangenlem 34627 subfacp1lem5 34640 erdsze2lem2 34660 pprodcnveq 35326 cocnv 37059 ltrncoidN 39465 trlcoabs2N 40059 trlcoat 40060 trlcone 40065 cdlemg46 40072 cdlemg47 40073 ltrnco4 40076 tgrpgrplem 40086 tendoplass 40120 cdlemi2 40156 cdlemk2 40169 cdlemk4 40171 cdlemk8 40175 cdlemk45 40284 cdlemk54 40295 cdlemk55a 40296 erngdvlem3 40327 erngdvlem3-rN 40335 tendocnv 40358 dvhvaddass 40434 dvhlveclem 40445 cdlemn8 40541 dihopelvalcpre 40585 dih1dimatlem0 40665 diophrw 41962 eldioph2 41965 mendring 42399 cortrcltrcl 42956 corclrtrcl 42957 cortrclrcl 42959 cotrclrtrcl 42960 cortrclrtrcl 42961 frege131d 42980 brcofffn 43247 brco3f1o 43249 neicvgnvo 43331 volicoff 45172 voliooicof 45173 ovolval4lem2 45827 isomushgr 46955 rngccatidALTV 47111 ringccatidALTV 47145 |
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