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Mirrors > Home > ILE Home > Th. List > coass | Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5037 | . 2 | |
2 | relco 5037 | . 2 | |
3 | excom 1642 | . . . 4 | |
4 | anass 398 | . . . . 5 | |
5 | 4 | 2exbii 1585 | . . . 4 |
6 | 3, 5 | bitr4i 186 | . . 3 |
7 | vex 2689 | . . . . . . 7 | |
8 | vex 2689 | . . . . . . 7 | |
9 | 7, 8 | brco 4710 | . . . . . 6 |
10 | 9 | anbi2i 452 | . . . . 5 |
11 | 10 | exbii 1584 | . . . 4 |
12 | vex 2689 | . . . . 5 | |
13 | 12, 8 | opelco 4711 | . . . 4 |
14 | exdistr 1881 | . . . 4 | |
15 | 11, 13, 14 | 3bitr4i 211 | . . 3 |
16 | vex 2689 | . . . . . . 7 | |
17 | 12, 16 | brco 4710 | . . . . . 6 |
18 | 17 | anbi1i 453 | . . . . 5 |
19 | 18 | exbii 1584 | . . . 4 |
20 | 12, 8 | opelco 4711 | . . . 4 |
21 | 19.41v 1874 | . . . . 5 | |
22 | 21 | exbii 1584 | . . . 4 |
23 | 19, 20, 22 | 3bitr4i 211 | . . 3 |
24 | 6, 15, 23 | 3bitr4i 211 | . 2 |
25 | 1, 2, 24 | eqrelriiv 4633 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 cop 3530 class class class wbr 3929 ccom 4543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-co 4548 |
This theorem is referenced by: funcoeqres 5398 fcof1o 5690 tposco 6172 mapen 6740 hashfacen 10579 |
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