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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 5878 | . 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∘ 𝐶) | |
2 | relco 5878 | . 2 ⊢ Rel (𝐴 ∘ (𝐵 ∘ 𝐶)) | |
3 | excom 2212 | . . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) | |
4 | anass 462 | . . . . 5 ⊢ (((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) | |
5 | 4 | 2exbii 1948 | . . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) |
6 | 3, 5 | bitr4i 270 | . . 3 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) |
7 | vex 3417 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
8 | vex 3417 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brco 5529 | . . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) |
10 | 9 | anbi2i 616 | . . . . 5 ⊢ ((𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) |
11 | 10 | exbii 1947 | . . . 4 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) |
12 | vex 3417 | . . . . 5 ⊢ 𝑥 ∈ V | |
13 | 12, 8 | opelco 5530 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦)) |
14 | exdistr 2053 | . . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) | |
15 | 11, 13, 14 | 3bitr4i 295 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) |
16 | vex 3417 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
17 | 12, 16 | brco 5529 | . . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤)) |
18 | 17 | anbi1i 617 | . . . . 5 ⊢ ((𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) |
19 | 18 | exbii 1947 | . . . 4 ⊢ (∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) |
20 | 12, 8 | opelco 5530 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦)) |
21 | 19.41v 2048 | . . . . 5 ⊢ (∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) | |
22 | 21 | exbii 1947 | . . . 4 ⊢ (∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) |
23 | 19, 20, 22 | 3bitr4i 295 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) |
24 | 6, 15, 23 | 3bitr4i 295 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 ∘ (𝐵 ∘ 𝐶))) |
25 | 1, 2, 24 | eqrelriiv 5452 | 1 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1656 ∃wex 1878 ∈ wcel 2164 〈cop 4405 class class class wbr 4875 ∘ ccom 5350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-co 5355 |
This theorem is referenced by: funcoeqres 6412 fcof1oinvd 6808 tposco 7653 mapen 8399 mapfien 8588 hashfacen 13534 relexpsucnnl 14156 relexpaddnn 14175 cofuass 16908 setccatid 17093 estrccatid 17131 frmdup3lem 17764 symggrp 18177 f1omvdco2 18225 symggen 18247 psgnunilem1 18270 gsumval3 18668 gsumzf1o 18673 gsumzmhm 18697 prds1 18975 psrass1lem 19745 pf1mpf 20083 pf1ind 20086 qtophmeo 21998 uniioombllem2 23756 cncombf 23831 motgrp 25862 pjsdi2i 29567 pjadj2coi 29614 pj3lem1 29616 pj3i 29618 fcoinver 29961 fmptco1f1o 29979 fcobij 30044 fcobijfs 30045 symgfcoeu 30386 reprpmtf1o 31249 derangenlem 31695 subfacp1lem5 31708 erdsze2lem2 31728 pprodcnveq 32524 cocnv 34058 ltrncoidN 36198 trlcoabs2N 36792 trlcoat 36793 trlcone 36798 cdlemg46 36805 cdlemg47 36806 ltrnco4 36809 tgrpgrplem 36819 tendoplass 36853 cdlemi2 36889 cdlemk2 36902 cdlemk4 36904 cdlemk8 36908 cdlemk45 37017 cdlemk54 37028 cdlemk55a 37029 erngdvlem3 37060 erngdvlem3-rN 37068 tendocnv 37091 dvhvaddass 37167 dvhlveclem 37178 cdlemn8 37274 dihopelvalcpre 37318 dih1dimatlem0 37398 diophrw 38161 eldioph2 38164 mendring 38600 cortrcltrcl 38868 corclrtrcl 38869 cortrclrcl 38871 cotrclrtrcl 38872 cortrclrtrcl 38873 frege131d 38892 brcofffn 39164 brco3f1o 39166 neicvgnvo 39248 volicoff 41000 voliooicof 41001 ovolval4lem2 41652 isomushgr 42558 rngccatidALTV 42850 ringccatidALTV 42913 |
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