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Theorem cotr2 14545
Description: Two ways of saying a relation is transitive. Special instance of cotr2g 14544. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr2.a dom 𝑅𝐴
cotr2.b (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
cotr2.c ran 𝑅𝐶
Assertion
Ref Expression
cotr2 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr2
StepHypRef Expression
1 cotr2.a . 2 dom 𝑅𝐴
2 incom 4120 . . 3 (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅)
3 cotr2.b . . 3 (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
42, 3eqsstrri 3941 . 2 (ran 𝑅 ∩ dom 𝑅) ⊆ 𝐵
5 cotr2.c . 2 ran 𝑅𝐶
61, 4, 5cotr2g 14544 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wral 3061  cin 3870  wss 3871   class class class wbr 5058  dom cdm 5556  ran crn 5557  ccom 5560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-br 5059  df-opab 5121  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567
This theorem is referenced by:  cotr3  14546
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