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Theorem resco 6114
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 5880 . 2 Rel ((𝐴𝐵) ↾ 𝐶)
2 relco 6108 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 vex 3412 . . . . . 6 𝑥 ∈ V
4 vex 3412 . . . . . 6 𝑦 ∈ V
53, 4brco 5739 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
65anbi2i 626 . . . 4 ((𝑥𝐶𝑥(𝐴𝐵)𝑦) ↔ (𝑥𝐶 ∧ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)))
7 19.42v 1962 . . . 4 (∃𝑧(𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐶 ∧ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)))
8 vex 3412 . . . . . . . 8 𝑧 ∈ V
98brresi 5860 . . . . . . 7 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐶𝑥𝐵𝑧))
109anbi1i 627 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐶𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦))
11 anass 472 . . . . . 6 (((𝑥𝐶𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)))
1210, 11bitr2i 279 . . . . 5 ((𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ (𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1312exbii 1855 . . . 4 (∃𝑧(𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
146, 7, 133bitr2i 302 . . 3 ((𝑥𝐶𝑥(𝐴𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
154brresi 5860 . . 3 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦 ↔ (𝑥𝐶𝑥(𝐴𝐵)𝑦))
163, 4brco 5739 . . 3 (𝑥(𝐴 ∘ (𝐵𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1714, 15, 163bitr4i 306 . 2 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦𝑥(𝐴 ∘ (𝐵𝐶))𝑦)
181, 2, 17eqbrriv 5661 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  wex 1787  wcel 2110   class class class wbr 5053  cres 5553  ccom 5555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-co 5560  df-res 5563
This theorem is referenced by:  cocnvcnv2  6122  coires1  6128  dftpos2  7985  canthp1lem2  10267  o1res  15121  gsumzaddlem  19306  tsmsf1o  23042  tsmsmhm  23043  mbfres  24541  hhssims  29355  symgcom  31071  cycpmconjslem1  31140  cycpmconjslem2  31141  erdsze2lem2  32879  cvmlift2lem9a  32978  ttrclco  33517  mbfresfi  35560  cocnv  35620  xrnres  36265  xrnres2  36266  xrnres3  36267  diophrw  40284  eldioph2  40287  mbfres2cn  43174  funcoressn  44208
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