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Theorem coass 4993
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 4973 . 2 Rel ((𝐴𝐵) ∘ 𝐶)
2 relco 4973 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 excom 1610 . . . 4 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑤𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
4 anass 396 . . . . 5 (((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
542exbii 1553 . . . 4 (∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
63, 5bitr4i 186 . . 3 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
7 vex 2644 . . . . . . 7 𝑧 ∈ V
8 vex 2644 . . . . . . 7 𝑦 ∈ V
97, 8brco 4648 . . . . . 6 (𝑧(𝐴𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦))
109anbi2i 448 . . . . 5 ((𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
1110exbii 1552 . . . 4 (∃𝑧(𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
12 vex 2644 . . . . 5 𝑥 ∈ V
1312, 8opelco 4649 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦))
14 exdistr 1846 . . . 4 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
1511, 13, 143bitr4i 211 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
16 vex 2644 . . . . . . 7 𝑤 ∈ V
1712, 16brco 4648 . . . . . 6 (𝑥(𝐵𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤))
1817anbi1i 449 . . . . 5 ((𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
1918exbii 1552 . . . 4 (∃𝑤(𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2012, 8opelco 4649 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)) ↔ ∃𝑤(𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦))
21 19.41v 1841 . . . . 5 (∃𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2221exbii 1552 . . . 4 (∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2319, 20, 223bitr4i 211 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)) ↔ ∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
246, 15, 233bitr4i 211 . 2 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)))
251, 2, 24eqrelriiv 4571 1 ((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1299  wex 1436  wcel 1448  cop 3477   class class class wbr 3875  ccom 4481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-co 4486
This theorem is referenced by:  funcoeqres  5332  fcof1o  5622  tposco  6102  mapen  6669  hashfacen  10420
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