Theorem cotrg 6127

Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6130 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6130. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2157. (Revised by BTernaryTau, 29-Dec-2024.)

Assertion

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

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

Proof of Theorem cotrg

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6126	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		ssrel3 5796	. . 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 3484	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3484	. . . . . . . 8 ⊢ 𝑧 ∈ V
6	4, 5	brco 5881	. . . . . . 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		breq2 5147	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑦𝐴𝑤))
12	11	anbi2d 630	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤)))
13		breq2 5147	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝑤))
14	12, 13	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝑤 → (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤) → 𝑥𝐶𝑤)))
15		breq2 5147	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑤))
16		breq1 5146	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑤𝐴𝑧))
17	15, 16	anbi12d 632	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧)))
18	17	imbi1d 341	. . . . 5 ⊢ (𝑦 = 𝑤 → (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧) → 𝑥𝐶𝑧)))
19	14, 18	alcomw 2044	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
20	10, 19	bitri 275	. . 3 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
21	20	albii 1819	. 2 ⊢ (∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
22	3, 21	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 3951   class class class wbr 5143   ∘ ccom 5689  Rel wrel 5690

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-co 5694

This theorem is referenced by:  cotr  6130  dffun2  6571  dffun2OLD  6572  cotr2g  15015