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Theorem cotrg 5665
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5666 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5666. (Revised by Richard Penner, 24-Dec-2019.)
Assertion
Ref Expression
cotrg ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cotrg
StepHypRef Expression
1 df-co 5275 . . . 4 (𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧)}
21relopabi 5401 . . 3 Rel (𝐴𝐵)
3 ssrel 5364 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶)))
42, 3ax-mp 5 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶))
5 vex 3343 . . . . . . . 8 𝑥 ∈ V
6 vex 3343 . . . . . . . 8 𝑧 ∈ V
75, 6opelco 5449 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
8 df-br 4805 . . . . . . . 8 (𝑥𝐶𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐶)
98bicomi 214 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝐶𝑥𝐶𝑧)
107, 9imbi12i 339 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11 19.23v 2020 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1210, 11bitr4i 267 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1312albii 1896 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14 alcom 2186 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1513, 14bitri 264 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1615albii 1896 . 2 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
174, 16bitri 264 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wex 1853  wcel 2139  wss 3715  cop 4327   class class class wbr 4804  ccom 5270  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-co 5275
This theorem is referenced by:  cotr  5666  cotr2g  13936
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