Theorem cotrgOLDOLD 6141

Description: Obsolete version of cotrg 6139 as of 19-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6142. (Revised by Richard Penner, 24-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cotrgOLDOLD	⊢ ((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 cotrgOLDOLD

Step	Hyp	Ref	Expression
1		relco 6138	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		ssrel 5806	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶)))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶))
4		vex 3492	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3492	. . . . . . . 8 ⊢ 𝑧 ∈ V
6	4, 5	opelco 5896	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
7		df-br 5167	. . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐶)
8	7	bicomi 224	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐶 ↔ 𝑥𝐶𝑧)
9	6, 8	imbi12i 350	. . . . . 6 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
10		19.23v 1941	. . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11	9, 10	bitr4i 278	. . . . 5 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
12	11	albii 1817	. . . 4 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
13		alcom 2160	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14	12, 13	bitri 275	. . 3 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
15	14	albii 1817	. 2 ⊢ (∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
16	3, 15	bitri 275	1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777   ∈ wcel 2108   ⊆ wss 3976  ⟨cop 4654   class class class wbr 5166   ∘ ccom 5704  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-co 5709

This theorem is referenced by: (None)