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Theorem cotr3 13565
 Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr3.a 𝐴 = dom 𝑅
cotr3.b 𝐵 = (𝐴𝐶)
cotr3.c 𝐶 = ran 𝑅
Assertion
Ref Expression
cotr3 ((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 cotr3
StepHypRef Expression
1 cotr3.a . . 3 𝐴 = dom 𝑅
21eqimss2i 3623 . 2 dom 𝑅𝐴
3 cotr3.b . . . 4 𝐵 = (𝐴𝐶)
4 cotr3.c . . . . 5 𝐶 = ran 𝑅
51, 4ineq12i 3774 . . . 4 (𝐴𝐶) = (dom 𝑅 ∩ ran 𝑅)
63, 5eqtri 2632 . . 3 𝐵 = (dom 𝑅 ∩ ran 𝑅)
76eqimss2i 3623 . 2 (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
84eqimss2i 3623 . 2 ran 𝑅𝐶
92, 7, 8cotr2 13564 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  ran crn 5039   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049 This theorem is referenced by: (None)
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