Theorem resco 6250

Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Assertion

Ref	Expression
resco	⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

Proof of Theorem resco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 6002	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶)
2		relco 6108	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶))
3		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	brco 5854	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6	5	anbi2i 635	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
7		19.42v 1986	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
8		vex 3457	. . . . . . . 8 ⊢ 𝑧 ∈ V
9	8	brresi 5985	. . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧))
10	9	anbi1i 636	. . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦))
11		anass 474	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
12	10, 11	bitr2i 279	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
13	12	exbii 1881	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
14	6, 7, 13	3bitr2i 302	. . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
15	4	brresi 5985	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦))
16	3, 4	brco 5854	. . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
17	14, 15, 16	3bitr4i 306	. 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦)
18	1, 2, 17	eqbrriv 5775	1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   class class class wbr 5107   ↾ cres 5661   ∘ ccom 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-co 5668  df-res 5671

This theorem is used by:  cocnvcnv2  6259  coires1  6265  dftpos2  8244  ttrclco  9700  canthp1lem2  10665  o1res  15649  gsumzaddlem  20049  tsmsf1o  24372  tsmsmhm  24373  mbfres  25873  hhssims  31741  symgcom  33510  cycpmconjslem1  33581  cycpmconjslem2  33582  erdsze2lem2  35770  cvmlift2lem9a  35869  mbfresfi  38402  cocnv  38462  xrnres  39160  xrnres2  39161  xrnres3  39162  diophrw  43591  eldioph2  43594  mbfres2cn  46773  funcoressn  47917  upgrimpthslem1  48810  tposrescnv  49792