Theorem resco 6237

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 5991	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶)
2		relco 6097	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶))
3		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3459	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	brco 5843	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6	5	anbi2i 632	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
7		19.42v 1974	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
8		vex 3459	. . . . . . . 8 ⊢ 𝑧 ∈ V
9	8	brresi 5974	. . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧))
10	9	anbi1i 633	. . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦))
11		anass 472	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
12	10, 11	bitr2i 278	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
13	12	exbii 1869	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
14	6, 7, 13	3bitr2i 301	. . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
15	4	brresi 5974	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦))
16	3, 4	brco 5843	. . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
17	14, 15, 16	3bitr4i 305	. 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦)
18	1, 2, 17	eqbrriv 5764	1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

Syntax hints:   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143   class class class wbr 5101   ↾ cres 5650   ∘ ccom 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-rel 5655  df-co 5657  df-res 5660

This theorem is referenced by:  cocnvcnv2  6246  coires1  6252  dftpos2  8223  ttrclco  9671  canthp1lem2  10622  o1res  15597  gsumzaddlem  19971  tsmsf1o  24212  tsmsmhm  24213  mbfres  25713  hhssims  31484  symgcom  33269  cycpmconjslem1  33340  cycpmconjslem2  33341  erdsze2lem2  35559  cvmlift2lem9a  35658  mbfresfi  38170  cocnv  38229  xrnres  38929  xrnres2  38930  xrnres3  38931  diophrw  43345  eldioph2  43348  mbfres2cn  46523  funcoressn  47627  upgrimpthslem1  48520  tposrescnv  49491