Theorem cotrg 5507

Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5508 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5508. (Revised by Richard Penner, 24-Dec-2019.)

Assertion

Ref	Expression
cotrg	⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cotrg

Step	Hyp	Ref	Expression
1		df-co 5123	. . . 4 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)}
2	1	relopabi 5245	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
3		ssrel 5207	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶)))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶))
5		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
6		vex 3203	. . . . . . . 8 ⊢ 𝑧 ∈ V
7	5, 6	opelco 5293	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
8		df-br 4654	. . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐶)
9	8	bicomi 214	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐶 ↔ 𝑥𝐶𝑧)
10	7, 9	imbi12i 340	. . . . . 6 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11		19.23v 1902	. . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
12	10, 11	bitr4i 267	. . . . 5 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
13	12	albii 1747	. . . 4 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14		alcom 2037	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
15	13, 14	bitri 264	. . 3 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
16	15	albii 1747	. 2 ⊢ (∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
17	4, 16	bitri 264	1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704   ∈ wcel 1990   ⊆ wss 3574  ⟨cop 4183   class class class wbr 4653   ∘ ccom 5118  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  cotr  5508  cotr2g  13715