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Theorem resco 4935
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 4741 . 2 Rel ((𝐴𝐵) ↾ 𝐶)
2 relco 4929 . 2 Rel (𝐴 ∘ (𝐵𝐶))
3 vex 2622 . . . . . 6 𝑥 ∈ V
4 vex 2622 . . . . . 6 𝑦 ∈ V
53, 4brco 4607 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
65anbi1i 446 . . . 4 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
7 19.41v 1830 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶))
8 an32 529 . . . . . 6 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
9 vex 2622 . . . . . . . 8 𝑧 ∈ V
109brres 4719 . . . . . . 7 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐵𝑧𝑥𝐶))
1110anbi1i 446 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑥𝐶) ∧ 𝑧𝐴𝑦))
128, 11bitr4i 185 . . . . 5 (((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ (𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1312exbii 1541 . . . 4 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∧ 𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
146, 7, 133bitr2i 206 . . 3 ((𝑥(𝐴𝐵)𝑦𝑥𝐶) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
154brres 4719 . . 3 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴𝐵)𝑦𝑥𝐶))
163, 4brco 4607 . . 3 (𝑥(𝐴 ∘ (𝐵𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
1714, 15, 163bitr4i 210 . 2 (𝑥((𝐴𝐵) ↾ 𝐶)𝑦𝑥(𝐴 ∘ (𝐵𝐶))𝑦)
181, 2, 17eqbrriv 4533 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wex 1426  wcel 1438   class class class wbr 3845  cres 4440  ccom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-co 4447  df-res 4450
This theorem is referenced by:  cocnvcnv2  4942  coires1  4948  relcoi1  4962  dftpos2  6026
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