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Theorem coass 5165
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 5145 . 2 Rel ((𝐴𝐵) ∘ 𝐶)
2 relco 5145 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 excom 1675 . . . 4 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑤𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
4 anass 401 . . . . 5 (((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
542exbii 1617 . . . 4 (∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
63, 5bitr4i 187 . . 3 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
7 vex 2755 . . . . . . 7 𝑧 ∈ V
8 vex 2755 . . . . . . 7 𝑦 ∈ V
97, 8brco 4816 . . . . . 6 (𝑧(𝐴𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦))
109anbi2i 457 . . . . 5 ((𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
1110exbii 1616 . . . 4 (∃𝑧(𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
12 vex 2755 . . . . 5 𝑥 ∈ V
1312, 8opelco 4817 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦))
14 exdistr 1921 . . . 4 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
1511, 13, 143bitr4i 212 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
16 vex 2755 . . . . . . 7 𝑤 ∈ V
1712, 16brco 4816 . . . . . 6 (𝑥(𝐵𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤))
1817anbi1i 458 . . . . 5 ((𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
1918exbii 1616 . . . 4 (∃𝑤(𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2012, 8opelco 4817 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)) ↔ ∃𝑤(𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦))
21 19.41v 1914 . . . . 5 (∃𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2221exbii 1616 . . . 4 (∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2319, 20, 223bitr4i 212 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)) ↔ ∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
246, 15, 233bitr4i 212 . 2 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)))
251, 2, 24eqrelriiv 4738 1 ((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wex 1503  wcel 2160  cop 3610   class class class wbr 4018  ccom 4648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-co 4653
This theorem is referenced by:  funcoeqres  5511  fcof1o  5811  tposco  6300  mapen  6874  hashfacen  10848
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