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Theorem resco 5151
Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
resco ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))

Proof of Theorem resco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4953 . 2 Rel ((𝐴𝐵) ↾ 𝐶)
2 relco 5145 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 vex 2755 . . . . . 6 𝑥 ∈ V
4 vex 2755 . . . . . 6 𝑦 ∈ V
53, 4brco 4816 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
65anbi1i 458 . . . 4 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
7 19.41v 1914 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
8 an32 562 . . . . . 6 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
9 vex 2755 . . . . . . . 8 𝑧 ∈ V
109brres 4931 . . . . . . 7 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐵𝑧𝑥𝐶))
1110anbi1i 458 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
128, 11bitr4i 187 . . . . 5 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1312exbii 1616 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
146, 7, 133bitr2i 208 . . 3 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
154brres 4931 . . 3 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴𝐵)𝑦𝑥𝐶))
163, 4brco 4816 . . 3 (𝑥(𝐴 ∘ (𝐵𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1714, 15, 163bitr4i 212 . 2 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦𝑥(𝐴 ∘ (𝐵𝐶))𝑦)
181, 2, 17eqbrriv 4739 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wex 1503  wcel 2160   class class class wbr 4018  cres 4646  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  df-res 4656
This theorem is referenced by:  cocnvcnv2  5158  coires1  5164  relcoi1  5178  dftpos2  6285
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