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Theorem cotrg 5964
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5965 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5965. (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 5557 . . . 4 (𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧)}
21relopabi 5687 . . 3 Rel (𝐴𝐵)
3 ssrel 5650 . . 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 3495 . . . . . . . 8 𝑥 ∈ V
6 vex 3495 . . . . . . . 8 𝑧 ∈ V
75, 6opelco 5735 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
8 df-br 5058 . . . . . . . 8 (𝑥𝐶𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐶)
98bicomi 225 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝐶𝑥𝐶𝑧)
107, 9imbi12i 352 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11 19.23v 1934 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1210, 11bitr4i 279 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1312albii 1811 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14 alcom 2153 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1513, 14bitri 276 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1615albii 1811 . 2 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
174, 16bitri 276 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wex 1771  wcel 2105  wss 3933  cop 4563   class class class wbr 5057  ccom 5552  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-co 5557
This theorem is referenced by:  cotr  5965  cotr2g  14324
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