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Theorem coass 4762
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 4742 . 2 Rel ((AB) ∘ 𝐶)
2 relco 4742 . 2 Rel (A ∘ (B𝐶))
3 excom 1532 . . . 4 (zw(x𝐶z (zBw wAy)) ↔ wz(x𝐶z (zBw wAy)))
4 anass 383 . . . . 5 (((x𝐶z zBw) wAy) ↔ (x𝐶z (zBw wAy)))
542exbii 1475 . . . 4 (wz((x𝐶z zBw) wAy) ↔ wz(x𝐶z (zBw wAy)))
63, 5bitr4i 176 . . 3 (zw(x𝐶z (zBw wAy)) ↔ wz((x𝐶z zBw) wAy))
7 vex 2534 . . . . . . 7 z V
8 vex 2534 . . . . . . 7 y V
97, 8brco 4429 . . . . . 6 (z(AB)yw(zBw wAy))
109anbi2i 433 . . . . 5 ((x𝐶z z(AB)y) ↔ (x𝐶z w(zBw wAy)))
1110exbii 1474 . . . 4 (z(x𝐶z z(AB)y) ↔ z(x𝐶z w(zBw wAy)))
12 vex 2534 . . . . 5 x V
1312, 8opelco 4430 . . . 4 (⟨x, y ((AB) ∘ 𝐶) ↔ z(x𝐶z z(AB)y))
14 exdistr 1765 . . . 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 2534 . . . . . . 7 w V
1712, 16brco 4429 . . . . . 6 (x(B𝐶)wz(x𝐶z zBw))
1817anbi1i 434 . . . . 5 ((x(B𝐶)w wAy) ↔ (z(x𝐶z zBw) wAy))
1918exbii 1474 . . . 4 (w(x(B𝐶)w wAy) ↔ w(z(x𝐶z zBw) wAy))
2012, 8opelco 4430 . . . 4 (⟨x, y (A ∘ (B𝐶)) ↔ w(x(B𝐶)w wAy))
21 19.41v 1760 . . . . 5 (z((x𝐶z zBw) wAy) ↔ (z(x𝐶z zBw) wAy))
2221exbii 1474 . . . 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 4357 1 ((AB) ∘ 𝐶) = (A ∘ (B𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226  wex 1358   wcel 1370  cop 3349   class class class wbr 3734  ccom 4272
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-co 4277
This theorem is referenced by:  funcoeqres  5078  fcof1o  5350  tposco  5808
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