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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 ((AB) ∘ 𝐶) = (A ∘ (B𝐶))

Proof of Theorem coass
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 4762 . 2 Rel ((AB) ∘ 𝐶)
2 relco 4762 . 2 Rel (A ∘ (B𝐶))
3 excom 1551 . . . 4 (zw(x𝐶z (zBw wAy)) ↔ wz(x𝐶z (zBw wAy)))
4 anass 381 . . . . 5 (((x𝐶z zBw) wAy) ↔ (x𝐶z (zBw wAy)))
542exbii 1494 . . . 4 (wz((x𝐶z zBw) wAy) ↔ wz(x𝐶z (zBw wAy)))
63, 5bitr4i 176 . . 3 (zw(x𝐶z (zBw wAy)) ↔ wz((x𝐶z zBw) wAy))
7 vex 2554 . . . . . . 7 z V
8 vex 2554 . . . . . . 7 y V
97, 8brco 4449 . . . . . 6 (z(AB)yw(zBw wAy))
109anbi2i 430 . . . . 5 ((x𝐶z z(AB)y) ↔ (x𝐶z w(zBw wAy)))
1110exbii 1493 . . . 4 (z(x𝐶z z(AB)y) ↔ z(x𝐶z w(zBw wAy)))
12 vex 2554 . . . . 5 x V
1312, 8opelco 4450 . . . 4 (⟨x, y ((AB) ∘ 𝐶) ↔ z(x𝐶z z(AB)y))
14 exdistr 1784 . . . 4 (zw(x𝐶z (zBw wAy)) ↔ z(x𝐶z w(zBw wAy)))
1511, 13, 143bitr4i 201 . . 3 (⟨x, y ((AB) ∘ 𝐶) ↔ zw(x𝐶z (zBw wAy)))
16 vex 2554 . . . . . . 7 w V
1712, 16brco 4449 . . . . . 6 (x(B𝐶)wz(x𝐶z zBw))
1817anbi1i 431 . . . . 5 ((x(B𝐶)w wAy) ↔ (z(x𝐶z zBw) wAy))
1918exbii 1493 . . . 4 (w(x(B𝐶)w wAy) ↔ w(z(x𝐶z zBw) wAy))
2012, 8opelco 4450 . . . 4 (⟨x, y (A ∘ (B𝐶)) ↔ w(x(B𝐶)w wAy))
21 19.41v 1779 . . . . 5 (z((x𝐶z zBw) wAy) ↔ (z(x𝐶z zBw) wAy))
2221exbii 1493 . . . 4 (wz((x𝐶z zBw) wAy) ↔ w(z(x𝐶z zBw) wAy))
2319, 20, 223bitr4i 201 . . 3 (⟨x, y (A ∘ (B𝐶)) ↔ wz((x𝐶z zBw) wAy))
246, 15, 233bitr4i 201 . 2 (⟨x, y ((AB) ∘ 𝐶) ↔ ⟨x, y (A ∘ (B𝐶)))
251, 2, 24eqrelriiv 4377 1 ((AB) ∘ 𝐶) = (A ∘ (B𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  cop 3370   class class class wbr 3755  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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