Theorem resco 5192

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 4992	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶)
2		relco 5186	. 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶))
3		vex 2776	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 2776	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	brco 4853	. . . . 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 1927	. . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶))
8		an32 562	. . . . . 6 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
9		vex 2776	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	brres 4970	. . . . . . 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 1629	. . . 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 4970	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶))
16	3, 4	brco 4853	. . 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 4774	1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177   class class class wbr 4047   ↾ cres 4681   ∘ ccom 4683

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-rel 4686  df-co 4688  df-res 4691

This theorem is referenced by:  cocnvcnv2  5199  coires1  5205  relcoi1  5219  dftpos2  6354