Theorem resco 5241

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 5041	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶)
2		relco 5235	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶))
3		vex 2805	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 2805	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	brco 4901	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6	5	anbi1i 458	. . . 4 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶))
7		19.41v 1951	. . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶))
8		an32 564	. . . . . 6 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
9		vex 2805	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	brres 5019	. . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶))
11	10	anbi1i 458	. . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
12	8, 11	bitr4i 187	. . . . 5 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
13	12	exbii 1653	. . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
14	6, 7, 13	3bitr2i 208	. . 3 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
15	4	brres 5019	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶))
16	3, 4	brco 4901	. . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
17	14, 15, 16	3bitr4i 212	. 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦)
18	1, 2, 17	eqbrriv 4821	1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202   class class class wbr 4088   ↾ cres 4727   ∘ ccom 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-co 4734  df-res 4737

This theorem is referenced by:  cocnvcnv2  5248  coires1  5254  relcoi1  5268  dftpos2  6427  gfsump1  16707