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Theorem resco 5895
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 5677 . 2 Rel ((𝐴𝐵) ↾ 𝐶)
2 relco 5889 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 vex 3401 . . . . . 6 𝑥 ∈ V
4 vex 3401 . . . . . 6 𝑦 ∈ V
53, 4brco 5540 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
65anbi2i 616 . . . 4 ((𝑥𝐶𝑥(𝐴𝐵)𝑦) ↔ (𝑥𝐶 ∧ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)))
7 19.42v 1996 . . . 4 (∃𝑧(𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐶 ∧ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)))
8 vex 3401 . . . . . . . 8 𝑧 ∈ V
98brresi 5653 . . . . . . 7 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐶𝑥𝐵𝑧))
109anbi1i 617 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐶𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦))
11 anass 462 . . . . . 6 (((𝑥𝐶𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)))
1210, 11bitr2i 268 . . . . 5 ((𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ (𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1312exbii 1892 . . . 4 (∃𝑧(𝑥𝐶 ∧ (𝑥𝐵𝑧𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
146, 7, 133bitr2i 291 . . 3 ((𝑥𝐶𝑥(𝐴𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
154brresi 5653 . . 3 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦 ↔ (𝑥𝐶𝑥(𝐴𝐵)𝑦))
163, 4brco 5540 . . 3 (𝑥(𝐴 ∘ (𝐵𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1714, 15, 163bitr4i 295 . 2 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦𝑥(𝐴 ∘ (𝐵𝐶))𝑦)
181, 2, 17eqbrriv 5464 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  wex 1823  wcel 2107   class class class wbr 4888  cres 5359  ccom 5361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-co 5366  df-res 5369
This theorem is referenced by:  cocnvcnv2  5903  coires1  5909  dftpos2  7653  canthp1lem2  9812  o1res  14708  gsumzaddlem  18718  tsmsf1o  22367  tsmsmhm  22368  mbfres  23859  hhssims  28721  erdsze2lem2  31793  cvmlift2lem9a  31892  mbfresfi  34090  cocnv  34154  xrnres  34797  xrnres2  34798  xrnres3  34799  diophrw  38296  eldioph2  38299  mbfres2cn  41115  funcoressn  42120
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