Theorem cotrg 5973

Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5974 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5974. (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 5566	. . . 4 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)}
2	1	relopabi 5696	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
3		ssrel 5659	. . 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 3499	. . . . . . . 8 ⊢ 𝑥 ∈ V
6		vex 3499	. . . . . . . 8 ⊢ 𝑧 ∈ V
7	5, 6	opelco 5744	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
8		df-br 5069	. . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐶)
9	8	bicomi 226	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐶 ↔ 𝑥𝐶𝑧)
10	7, 9	imbi12i 353	. . . . . 6 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11		19.23v 1943	. . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
12	10, 11	bitr4i 280	. . . . 5 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
13	12	albii 1820	. . . 4 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14		alcom 2163	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
15	13, 14	bitri 277	. . 3 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
16	15	albii 1820	. 2 ⊢ (∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
17	4, 16	bitri 277	1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780   ∈ wcel 2114   ⊆ wss 3938  ⟨cop 4575   class class class wbr 5068   ∘ ccom 5561  Rel wrel 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-co 5566

This theorem is referenced by:  cotr  5974  cotr2g  14338