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Theorem cotrg 6109
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6112 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6112. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2155. (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 6108 . . 3 Rel (𝐴𝐵)
2 ssrel3 5787 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧)))
31, 2ax-mp 5 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧))
4 vex 3479 . . . . . . . 8 𝑥 ∈ V
5 vex 3479 . . . . . . . 8 𝑧 ∈ V
64, 5brco 5871 . . . . . . 7 (𝑥(𝐴𝐵)𝑧 ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
76imbi1i 350 . . . . . 6 ((𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
8 19.23v 1946 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
97, 8bitr4i 278 . . . . 5 ((𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
109albii 1822 . . . 4 (∀𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11 breq2 5153 . . . . . . 7 (𝑧 = 𝑤 → (𝑦𝐴𝑧𝑦𝐴𝑤))
1211anbi2d 630 . . . . . 6 (𝑧 = 𝑤 → ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦𝑦𝐴𝑤)))
13 breq2 5153 . . . . . 6 (𝑧 = 𝑤 → (𝑥𝐶𝑧𝑥𝐶𝑤))
1412, 13imbi12d 345 . . . . 5 (𝑧 = 𝑤 → (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑦𝑦𝐴𝑤) → 𝑥𝐶𝑤)))
15 breq2 5153 . . . . . . 7 (𝑦 = 𝑤 → (𝑥𝐵𝑦𝑥𝐵𝑤))
16 breq1 5152 . . . . . . 7 (𝑦 = 𝑤 → (𝑦𝐴𝑧𝑤𝐴𝑧))
1715, 16anbi12d 632 . . . . . 6 (𝑦 = 𝑤 → ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑥𝐵𝑤𝑤𝐴𝑧)))
1817imbi1d 342 . . . . 5 (𝑦 = 𝑤 → (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑤𝑤𝐴𝑧) → 𝑥𝐶𝑧)))
1914, 18alcomw 2048 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
2010, 19bitri 275 . . 3 (∀𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
2120albii 1822 . 2 (∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
223, 21bitri 275 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wss 3949   class class class wbr 5149  ccom 5681  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-co 5686
This theorem is referenced by:  cotr  6112  dffun2  6554  dffun2OLD  6555  cotr2g  14923
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