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Theorem cotr2 13937
Description: Two ways of saying a relation is transitive. Special instance of cotr2g 13936. (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 3948 . . 3 (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅)
3 cotr2.b . . 3 (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
42, 3eqsstr3i 3777 . 2 (ran 𝑅 ∩ dom 𝑅) ⊆ 𝐵
5 cotr2.c . 2 ran 𝑅𝐶
61, 4, 5cotr2g 13936 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wral 3050  cin 3714  wss 3715   class class class wbr 4804  dom cdm 5266  ran crn 5267  ccom 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277
This theorem is referenced by:  cotr3  13938
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