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Theorem resco 6250
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 6011 . 2 Rel ((𝐴𝐵) ↾ 𝐶)
2 relco 6108 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 vex 3479 . . . . . 6 𝑥 ∈ V
4 vex 3479 . . . . . 6 𝑦 ∈ V
53, 4brco 5871 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
65anbi2i 624 . . . 4 ((𝑥𝐶𝑥(𝐴𝐵)𝑦) ↔ (𝑥𝐶 ∧ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)))
7 19.42v 1958 . . . 4 (∃𝑧(𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐶 ∧ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)))
8 vex 3479 . . . . . . . 8 𝑧 ∈ V
98brresi 5991 . . . . . . 7 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐶𝑥𝐵𝑧))
109anbi1i 625 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐶𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦))
11 anass 470 . . . . . 6 (((𝑥𝐶𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)))
1210, 11bitr2i 276 . . . . 5 ((𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ (𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1312exbii 1851 . . . 4 (∃𝑧(𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
146, 7, 133bitr2i 299 . . 3 ((𝑥𝐶𝑥(𝐴𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
154brresi 5991 . . 3 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦 ↔ (𝑥𝐶𝑥(𝐴𝐵)𝑦))
163, 4brco 5871 . . 3 (𝑥(𝐴 ∘ (𝐵𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1714, 15, 163bitr4i 303 . 2 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦𝑥(𝐴 ∘ (𝐵𝐶))𝑦)
181, 2, 17eqbrriv 5792 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107   class class class wbr 5149  cres 5679  ccom 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-co 5686  df-res 5689
This theorem is referenced by:  cocnvcnv2  6258  coires1  6264  dftpos2  8228  ttrclco  9713  canthp1lem2  10648  o1res  15504  gsumzaddlem  19789  tsmsf1o  23649  tsmsmhm  23650  mbfres  25161  hhssims  30527  symgcom  32244  cycpmconjslem1  32313  cycpmconjslem2  32314  erdsze2lem2  34195  cvmlift2lem9a  34294  mbfresfi  36534  cocnv  36593  xrnres  37272  xrnres2  37273  xrnres3  37274  diophrw  41497  eldioph2  41500  mbfres2cn  44674  funcoressn  45752
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