Theorem resco 5115

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 4919	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶)
2		relco 5109	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶))
3		vex 2733	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 2733	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	brco 4782	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6	5	anbi1i 455	. . . 4 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶))
7		19.41v 1895	. . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶))
8		an32 557	. . . . . 6 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
9		vex 2733	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	brres 4897	. . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶))
11	10	anbi1i 455	. . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
12	8, 11	bitr4i 186	. . . . 5 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
13	12	exbii 1598	. . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
14	6, 7, 13	3bitr2i 207	. . 3 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
15	4	brres 4897	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶))
16	3, 4	brco 4782	. . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
17	14, 15, 16	3bitr4i 211	. 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦)
18	1, 2, 17	eqbrriv 4706	1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

Syntax hints:   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141   class class class wbr 3989   ↾ cres 4613   ∘ ccom 4615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-co 4620  df-res 4623

This theorem is referenced by:  cocnvcnv2  5122  coires1  5128  relcoi1  5142  dftpos2  6240