Theorem cotrgOLDOLD 6085

Description: Obsolete version of cotrg 6083 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 6086. (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 6082	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		ssrel 5748	. . 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 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3454	. . . . . . . 8 ⊢ 𝑧 ∈ V
6	4, 5	opelco 5838	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
7		df-br 5111	. . . . . . . 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 1942	. . . . . 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 1819	. . . 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 1819	. 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 1538  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3917  ⟨cop 4598   class class class wbr 5110   ∘ ccom 5645  Rel wrel 5646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-co 5650

This theorem is referenced by: (None)