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Theorem cotrg 6108
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6111 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6111. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.)
Assertion
Ref Expression
cotrg ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cotrg
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 relco 6107 . . 3 Rel (𝐴𝐵)
2 ssrel3 5786 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧)))
31, 2ax-mp 5 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧))
4 vex 3478 . . . . . . . 8 𝑥 ∈ V
5 vex 3478 . . . . . . . 8 𝑧 ∈ V
64, 5brco 5870 . . . . . . 7 (𝑥(𝐴𝐵)𝑧 ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
76imbi1i 349 . . . . . 6 ((𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
8 19.23v 1945 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
97, 8bitr4i 277 . . . . 5 ((𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
109albii 1821 . . . 4 (∀𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11 breq2 5152 . . . . . . 7 (𝑧 = 𝑤 → (𝑦𝐴𝑧𝑦𝐴𝑤))
1211anbi2d 629 . . . . . 6 (𝑧 = 𝑤 → ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦𝑦𝐴𝑤)))
13 breq2 5152 . . . . . 6 (𝑧 = 𝑤 → (𝑥𝐶𝑧𝑥𝐶𝑤))
1412, 13imbi12d 344 . . . . 5 (𝑧 = 𝑤 → (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑦𝑦𝐴𝑤) → 𝑥𝐶𝑤)))
15 breq2 5152 . . . . . . 7 (𝑦 = 𝑤 → (𝑥𝐵𝑦𝑥𝐵𝑤))
16 breq1 5151 . . . . . . 7 (𝑦 = 𝑤 → (𝑦𝐴𝑧𝑤𝐴𝑧))
1715, 16anbi12d 631 . . . . . 6 (𝑦 = 𝑤 → ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑥𝐵𝑤𝑤𝐴𝑧)))
1817imbi1d 341 . . . . 5 (𝑦 = 𝑤 → (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑤𝑤𝐴𝑧) → 𝑥𝐶𝑧)))
1914, 18alcomw 2047 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
2010, 19bitri 274 . . 3 (∀𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
2120albii 1821 . 2 (∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
223, 21bitri 274 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wex 1781  wss 3948   class class class wbr 5148  ccom 5680  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-co 5685
This theorem is referenced by:  cotr  6111  dffun2  6553  dffun2OLD  6554  cotr2g  14922
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