Theorem resco 6215

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 5971	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶)
2		relco 6074	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶))
3		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3434	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	brco 5826	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6	5	anbi2i 624	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
7		19.42v 1955	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
8		vex 3434	. . . . . . . 8 ⊢ 𝑧 ∈ V
9	8	brresi 5954	. . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧))
10	9	anbi1i 625	. . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦))
11		anass 468	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)))
12	10, 11	bitr2i 276	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
13	12	exbii 1850	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
14	6, 7, 13	3bitr2i 299	. . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
15	4	brresi 5954	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦))
16	3, 4	brco 5826	. . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
17	14, 15, 16	3bitr4i 303	. 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦)
18	1, 2, 17	eqbrriv 5747	1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   class class class wbr 5086   ↾ cres 5633   ∘ ccom 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5637  df-rel 5638  df-co 5640  df-res 5643

This theorem is referenced by:  cocnvcnv2  6224  coires1  6230  dftpos2  8193  ttrclco  9639  canthp1lem2  10576  o1res  15522  gsumzaddlem  19896  tsmsf1o  24110  tsmsmhm  24111  mbfres  25611  hhssims  31345  symgcom  33144  cycpmconjslem1  33215  cycpmconjslem2  33216  erdsze2lem2  35386  cvmlift2lem9a  35485  mbfresfi  37987  cocnv  38046  xrnres  38746  xrnres2  38747  xrnres3  38748  diophrw  43191  eldioph2  43194  mbfres2cn  46386  funcoressn  47484  upgrimpthslem1  48377  tposrescnv  49348