Theorem cotrg 6139

Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6142 for the main application. (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 shortened by SN, 19-Dec-2024.) Avoid ax-11 2158. (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 6138	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		ssrel3 5810	. . 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 3492	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3492	. . . . . . . 8 ⊢ 𝑧 ∈ V
6	4, 5	brco 5895	. . . . . . 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 1941	. . . . . 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 1817	. . . 4 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11		breq2 5170	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑦𝐴𝑤))
12	11	anbi2d 629	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤)))
13		breq2 5170	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝑤))
14	12, 13	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝑤 → (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤) → 𝑥𝐶𝑤)))
15		breq2 5170	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑤))
16		breq1 5169	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑤𝐴𝑧))
17	15, 16	anbi12d 631	. . . . . 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 1817	. 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 1535  ∃wex 1777   ⊆ wss 3976   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-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:  cotr  6142  dffun2  6583  dffun2OLD  6584  cotr2g  15025