Theorem resco 5556

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 5346	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶)
2		relco 5550	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶))
3		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3176	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	brco 5214	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6	5	anbi1i 727	. . . 4 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶))
7		19.41v 1901	. . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶))
8		an32 835	. . . . . 6 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
9		vex 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	brres 5323	. . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶))
11	10	anbi1i 727	. . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
12	8, 11	bitr4i 266	. . . . 5 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
13	12	exbii 1764	. . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
14	6, 7, 13	3bitr2i 287	. . 3 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
15	4	brres 5323	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶))
16	3, 4	brco 5214	. . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
17	14, 15, 16	3bitr4i 291	. 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦)
18	1, 2, 17	eqbrriv 5138	1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583   ↾ cres 5040   ∘ ccom 5042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-co 5047  df-res 5050

This theorem is referenced by:  cocnvcnv2  5564  coires1  5570  dftpos2  7256  canthp1lem2  9354  o1res  14139  gsumzaddlem  18144  tsmsf1o  21758  tsmsmhm  21759  mbfres  23217  hhssims  27516  erdsze2lem2  30440  cvmlift2lem9a  30539  mbfresfi  32626  cocnv  32690  diophrw  36340  eldioph2  36343  mbfres2cn  38850  funcoressn  39856