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Theorem cotrgOLDOLD 6132
Description: Obsolete version of cotrg 6130 as of 19-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6133. (Revised by Richard Penner, 24-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cotrgOLDOLD ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cotrgOLDOLD
StepHypRef Expression
1 relco 6129 . . 3 Rel (𝐴𝐵)
2 ssrel 5795 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶)))
31, 2ax-mp 5 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶))
4 vex 3482 . . . . . . . 8 𝑥 ∈ V
5 vex 3482 . . . . . . . 8 𝑧 ∈ V
64, 5opelco 5885 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
7 df-br 5149 . . . . . . . 8 (𝑥𝐶𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐶)
87bicomi 224 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝐶𝑥𝐶𝑧)
96, 8imbi12i 350 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
10 19.23v 1940 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
119, 10bitr4i 278 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1211albii 1816 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
13 alcom 2157 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1412, 13bitri 275 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1514albii 1816 . 2 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
163, 15bitri 275 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  wcel 2106  wss 3963  cop 4637   class class class wbr 5148  ccom 5693  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-co 5698
This theorem is referenced by: (None)
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