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Theorem cotrg 5968
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5969 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5969. (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 5562 . . . 4 (𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧)}
21relopabi 5692 . . 3 Rel (𝐴𝐵)
3 ssrel 5655 . . 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 3502 . . . . . . . 8 𝑥 ∈ V
6 vex 3502 . . . . . . . 8 𝑧 ∈ V
75, 6opelco 5740 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
8 df-br 5063 . . . . . . . 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 1936 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1210, 11bitr4i 279 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1312albii 1813 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14 alcom 2155 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1513, 14bitri 276 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1615albii 1813 . 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 1528  wex 1773  wcel 2107  wss 3939  cop 4569   class class class wbr 5062  ccom 5557  Rel wrel 5558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-co 5562
This theorem is referenced by:  cotr  5969  cotr2g  14329
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