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Theorem coass 4839
 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 4819 . 2 Rel ((𝐴𝐵) ∘ 𝐶)
2 relco 4819 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 excom 1554 . . . 4 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑤𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
4 anass 381 . . . . 5 (((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
542exbii 1497 . . . 4 (∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤𝑧(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
63, 5bitr4i 176 . . 3 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
7 vex 2560 . . . . . . 7 𝑧 ∈ V
8 vex 2560 . . . . . . 7 𝑦 ∈ V
97, 8brco 4506 . . . . . 6 (𝑧(𝐴𝐵)𝑦 ↔ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦))
109anbi2i 430 . . . . 5 ((𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦) ↔ (𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
1110exbii 1496 . . . 4 (∃𝑧(𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
12 vex 2560 . . . . 5 𝑥 ∈ V
1312, 8opelco 4507 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ∃𝑧(𝑥𝐶𝑧𝑧(𝐴𝐵)𝑦))
14 exdistr 1787 . . . 4 (∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)) ↔ ∃𝑧(𝑥𝐶𝑧 ∧ ∃𝑤(𝑧𝐵𝑤𝑤𝐴𝑦)))
1511, 13, 143bitr4i 201 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ∃𝑧𝑤(𝑥𝐶𝑧 ∧ (𝑧𝐵𝑤𝑤𝐴𝑦)))
16 vex 2560 . . . . . . 7 𝑤 ∈ V
1712, 16brco 4506 . . . . . 6 (𝑥(𝐵𝐶)𝑤 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤))
1817anbi1i 431 . . . . 5 ((𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
1918exbii 1496 . . . 4 (∃𝑤(𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2012, 8opelco 4507 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)) ↔ ∃𝑤(𝑥(𝐵𝐶)𝑤𝑤𝐴𝑦))
21 19.41v 1782 . . . . 5 (∃𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2221exbii 1496 . . . 4 (∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦) ↔ ∃𝑤(∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
2319, 20, 223bitr4i 201 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)) ↔ ∃𝑤𝑧((𝑥𝐶𝑧𝑧𝐵𝑤) ∧ 𝑤𝐴𝑦))
246, 15, 233bitr4i 201 . 2 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐵) ∘ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵𝐶)))
251, 2, 24eqrelriiv 4434 1 ((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764   ∘ ccom 4349 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-co 4354 This theorem is referenced by:  funcoeqres  5157  fcof1o  5429  tposco  5890
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