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Theorem resco 5783
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 5567 . 2 Rel ((𝐴𝐵) ↾ 𝐶)
2 relco 5777 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 vex 3354 . . . . . 6 𝑥 ∈ V
4 vex 3354 . . . . . 6 𝑦 ∈ V
53, 4brco 5431 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
65anbi1i 610 . . . 4 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
7 19.41v 2029 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
8 an32 625 . . . . . 6 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
9 vex 3354 . . . . . . . 8 𝑧 ∈ V
109brres 5543 . . . . . . 7 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐵𝑧𝑥𝐶))
1110anbi1i 610 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
128, 11bitr4i 267 . . . . 5 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1312exbii 1924 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
146, 7, 133bitr2i 288 . . 3 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
154brres 5543 . . 3 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴𝐵)𝑦𝑥𝐶))
163, 4brco 5431 . . 3 (𝑥(𝐴 ∘ (𝐵𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1714, 15, 163bitr4i 292 . 2 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦𝑥(𝐴 ∘ (𝐵𝐶))𝑦)
181, 2, 17eqbrriv 5355 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145   class class class wbr 4786  cres 5251  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-co 5258  df-res 5261
This theorem is referenced by:  cocnvcnv2  5791  coires1  5797  dftpos2  7521  canthp1lem2  9677  o1res  14499  gsumzaddlem  18528  tsmsf1o  22168  tsmsmhm  22169  mbfres  23631  hhssims  28472  erdsze2lem2  31524  cvmlift2lem9a  31623  mbfresfi  33788  cocnv  33852  xrnres  34502  xrnres2  34503  xrnres3  34504  diophrw  37848  eldioph2  37851  mbfres2cn  40691  funcoressn  41727
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