Theorem cotrg 6113

Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6114 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6114. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2195. (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 6112	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		ssrel3 5774	. . 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 3461	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3461	. . . . . . . 8 ⊢ 𝑧 ∈ V
6	4, 5	brco 5858	. . . . . . 7 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑧 ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
7	6	imbi1i 352	. . . . . 6 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
8		19.23v 1975	. . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
9	7, 8	bitr4i 281	. . . . 5 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
10	9	albii 1852	. . . 4 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11		breq2 5115	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑦𝐴𝑤))
12	11	anbi2d 642	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤)))
13		breq2 5115	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝑤))
14	12, 13	imbi12d 347	. . . . 5 ⊢ (𝑧 = 𝑤 → (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤) → 𝑥𝐶𝑤)))
15		breq2 5115	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑤))
16		breq1 5114	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑤𝐴𝑧))
17	15, 16	anbi12d 644	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧)))
18	17	imbi1d 344	. . . . 5 ⊢ (𝑦 = 𝑤 → (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧) → 𝑥𝐶𝑧)))
19	14, 18	alcomw 2078	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
20	10, 19	bitri 278	. . 3 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
21	20	albii 1852	. 2 ⊢ (∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
22	3, 21	bitri 278	1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ⊆ wss 3906   class class class wbr 5111   ∘ ccom 5667  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-co 5672

This theorem is used by:  cotr  6114  dffun2  6550  cotr2g  15039