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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 6148 | . 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∘ 𝐶) | |
2 | relco 6148 | . 2 ⊢ Rel (𝐴 ∘ (𝐵 ∘ 𝐶)) | |
3 | excom 2162 | . . . 4 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦))) | |
4 | anass 469 | . . . . 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 277 | . . 3 ⊢ (∃𝑧∃𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) ↔ ∃𝑤∃𝑧((𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦)) |
7 | vex 3436 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
8 | vex 3436 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brco 5779 | . . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤 ∧ 𝑤𝐴𝑦)) |
10 | 9 | anbi2i 623 | . . . . 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 3436 | . . . . 5 ⊢ 𝑥 ∈ V | |
13 | 12, 8 | opelco 5780 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 ∘ 𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧(𝐴 ∘ 𝐵)𝑦)) |
14 | exdistr 1958 | . . . 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 3436 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
17 | 12, 16 | brco 5779 | . . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑤)) |
18 | 17 | anbi1i 624 | . . . . 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 5780 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ (𝐵 ∘ 𝐶)) ↔ ∃𝑤(𝑥(𝐵 ∘ 𝐶)𝑤 ∧ 𝑤𝐴𝑦)) |
21 | 19.41v 1953 | . . . . 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 5700 | 1 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 〈cop 4567 class class class wbr 5074 ∘ ccom 5593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-co 5598 |
This theorem is referenced by: funcoeqres 6747 fcof1oinvd 7165 tposco 8073 mapen 8928 mapfien 9167 hashfacen 14166 hashfacenOLD 14167 relexpsucnnl 14741 relexpaddnn 14762 cofuass 17604 setccatid 17799 estrccatid 17848 frmdup3lem 18505 symggrplem 18523 f1omvdco2 19056 symggen 19078 psgnunilem1 19101 gsumval3 19508 gsumzf1o 19513 gsumzmhm 19538 prds1 19853 psrass1lemOLD 21143 psrass1lem 21146 pf1mpf 21518 pf1ind 21521 qtophmeo 22968 uniioombllem2 24747 cncombf 24822 motgrp 26904 pjsdi2i 30519 pjadj2coi 30566 pj3lem1 30568 pj3i 30570 fcoinver 30946 fmptco1f1o 30968 fcobij 31057 fcobijfs 31058 symgfcoeu 31351 pmtrcnel2 31359 cycpmconjv 31409 cycpmconjslem1 31421 cycpmconjs 31423 cyc3conja 31424 reprpmtf1o 32606 derangenlem 33133 subfacp1lem5 33146 erdsze2lem2 33166 pprodcnveq 34185 cocnv 35883 ltrncoidN 38142 trlcoabs2N 38736 trlcoat 38737 trlcone 38742 cdlemg46 38749 cdlemg47 38750 ltrnco4 38753 tgrpgrplem 38763 tendoplass 38797 cdlemi2 38833 cdlemk2 38846 cdlemk4 38848 cdlemk8 38852 cdlemk45 38961 cdlemk54 38972 cdlemk55a 38973 erngdvlem3 39004 erngdvlem3-rN 39012 tendocnv 39035 dvhvaddass 39111 dvhlveclem 39122 cdlemn8 39218 dihopelvalcpre 39262 dih1dimatlem0 39342 diophrw 40581 eldioph2 40584 mendring 41017 cortrcltrcl 41348 corclrtrcl 41349 cortrclrcl 41351 cotrclrtrcl 41352 cortrclrtrcl 41353 frege131d 41372 brcofffn 41641 brco3f1o 41643 neicvgnvo 41725 volicoff 43536 voliooicof 43537 ovolval4lem2 44188 isomushgr 45278 rngccatidALTV 45547 ringccatidALTV 45610 |
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