MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cotrgOLDOLD Structured version   Visualization version   GIF version

Theorem cotrgOLDOLD 6129
Description: Obsolete version of cotrg 6127 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 6130. (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 6126 . . 3 Rel (𝐴𝐵)
2 ssrel 5792 . . 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 3484 . . . . . . . 8 𝑥 ∈ V
5 vex 3484 . . . . . . . 8 𝑧 ∈ V
64, 5opelco 5882 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
7 df-br 5144 . . . . . . . 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 1942 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
119, 10bitr4i 278 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1211albii 1819 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
13 alcom 2159 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1412, 13bitri 275 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1514albii 1819 . 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 1538  wex 1779  wcel 2108  wss 3951  cop 4632   class class class wbr 5143  ccom 5689  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-co 5694
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator