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Theorem coass 4782
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.)
Assertion
Ref Expression
coass  o.  o.  C  o.  o.  C

Proof of Theorem coass
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 4762 . 2  Rel  o.  o.  C
2 relco 4762 . 2  Rel  o.  o.  C
3 excom 1551 . . . 4  C  C
4 anass 381 . . . . 5  C  C
542exbii 1494 . . . 4  C  C
63, 5bitr4i 176 . . 3  C  C
7 vex 2554 . . . . . . 7 
_V
8 vex 2554 . . . . . . 7 
_V
97, 8brco 4449 . . . . . 6  o.
109anbi2i 430 . . . . 5  C  o.  C
1110exbii 1493 . . . 4  C  o.  C
12 vex 2554 . . . . 5 
_V
1312, 8opelco 4450 . . . 4  <. ,  >.  o.  o.  C  C  o.
14 exdistr 1784 . . . 4  C  C
1511, 13, 143bitr4i 201 . . 3  <. ,  >.  o.  o.  C  C
16 vex 2554 . . . . . . 7 
_V
1712, 16brco 4449 . . . . . 6  o.  C  C
1817anbi1i 431 . . . . 5  o.  C  C
1918exbii 1493 . . . 4  o.  C  C
2012, 8opelco 4450 . . . 4  <. ,  >.  o.  o.  C  o.  C
21 19.41v 1779 . . . . 5  C  C
2221exbii 1493 . . . 4  C  C
2319, 20, 223bitr4i 201 . . 3  <. ,  >.  o.  o.  C  C
246, 15, 233bitr4i 201 . 2  <. ,  >.  o.  o.  C  <. ,  >.  o.  o.  C
251, 2, 24eqrelriiv 4377 1  o.  o.  C  o.  o.  C
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242  wex 1378   wcel 1390   <.cop 3370   class class class wbr 3755    o. ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-co 4297
This theorem is referenced by:  funcoeqres  5100  fcof1o  5372  tposco  5831
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