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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5045 |
. 2
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2 | relco 5045 |
. 2
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3 | excom 1643 |
. . . 4
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4 | anass 399 |
. . . . 5
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5 | 4 | 2exbii 1586 |
. . . 4
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6 | 3, 5 | bitr4i 186 |
. . 3
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7 | vex 2692 |
. . . . . . 7
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8 | vex 2692 |
. . . . . . 7
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9 | 7, 8 | brco 4718 |
. . . . . 6
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10 | 9 | anbi2i 453 |
. . . . 5
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11 | 10 | exbii 1585 |
. . . 4
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12 | vex 2692 |
. . . . 5
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13 | 12, 8 | opelco 4719 |
. . . 4
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14 | exdistr 1882 |
. . . 4
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15 | 11, 13, 14 | 3bitr4i 211 |
. . 3
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16 | vex 2692 |
. . . . . . 7
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17 | 12, 16 | brco 4718 |
. . . . . 6
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18 | 17 | anbi1i 454 |
. . . . 5
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19 | 18 | exbii 1585 |
. . . 4
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20 | 12, 8 | opelco 4719 |
. . . 4
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21 | 19.41v 1875 |
. . . . 5
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22 | 21 | exbii 1585 |
. . . 4
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23 | 19, 20, 22 | 3bitr4i 211 |
. . 3
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24 | 6, 15, 23 | 3bitr4i 211 |
. 2
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25 | 1, 2, 24 | eqrelriiv 4641 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-co 4556 |
This theorem is referenced by: funcoeqres 5406 fcof1o 5698 tposco 6180 mapen 6748 hashfacen 10611 |
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