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Theorem resco 4852
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 4666 . 2 Rel ((𝐴𝐵) ↾ 𝐶)
2 relco 4846 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 vex 2577 . . . . . 6 𝑥 ∈ V
4 vex 2577 . . . . . 6 𝑦 ∈ V
53, 4brco 4533 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
65anbi1i 439 . . . 4 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
7 19.41v 1798 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
8 an32 504 . . . . . 6 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
9 vex 2577 . . . . . . . 8 𝑧 ∈ V
109brres 4645 . . . . . . 7 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐵𝑧𝑥𝐶))
1110anbi1i 439 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
128, 11bitr4i 180 . . . . 5 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1312exbii 1512 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
146, 7, 133bitr2i 201 . . 3 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
154brres 4645 . . 3 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴𝐵)𝑦𝑥𝐶))
163, 4brco 4533 . . 3 (𝑥(𝐴 ∘ (𝐵𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1714, 15, 163bitr4i 205 . 2 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦𝑥(𝐴 ∘ (𝐵𝐶))𝑦)
181, 2, 17eqbrriv 4462 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409   class class class wbr 3791  cres 4374  ccom 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-co 4381  df-res 4384
This theorem is referenced by:  cocnvcnv2  4859  coires1  4865  relcoi1  4876  dftpos2  5906
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