Theorem cotrgOLD 6081

Description: Obsolete version of cotrg 6080 as of 29-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6083. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cotrgOLD	⊢ ((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 cotrgOLD

Step	Hyp	Ref	Expression
1		relco 6079	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		ssrel3 5749	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧)))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧))
4		vex 3451	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3451	. . . . . . . 8 ⊢ 𝑧 ∈ V
6	4, 5	brco 5834	. . . . . . 7 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑧 ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
7	6	imbi1i 349	. . . . . 6 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
8		19.23v 1942	. . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
9	7, 8	bitr4i 278	. . . . 5 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
10	9	albii 1819	. . . 4 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11		alcom 2160	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
12	10, 11	bitri 275	. . 3 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
13	12	albii 1819	. 2 ⊢ (∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14	3, 13	bitri 275	1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779   ⊆ wss 3914   class class class wbr 5107   ∘ ccom 5642  Rel wrel 5643

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-co 5647

This theorem is referenced by: (None)