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Theorem cotr2 13090
Description: Two ways of saying a relation is transitive. Special instance of cotr2g 13089. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr2.a  |-  dom  R  C_  A
cotr2.b  |-  ( dom 
R  i^i  ran  R ) 
C_  B
cotr2.c  |-  ran  R  C_  C
Assertion
Ref Expression
cotr2  |-  ( ( R  o.  R ) 
C_  R  <->  A. x  e.  A  A. y  e.  B  A. z  e.  C  ( (
x R y  /\  y R z )  ->  x R z ) )
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  R  C_  A
2 incom 3637 . . 3  |-  ( dom 
R  i^i  ran  R )  =  ( ran  R  i^i  dom  R )
3 cotr2.b . . 3  |-  ( dom 
R  i^i  ran  R ) 
C_  B
42, 3eqsstr3i 3475 . 2  |-  ( ran 
R  i^i  dom  R ) 
C_  B
5 cotr2.c . 2  |-  ran  R  C_  C
61, 4, 5cotr2g 13089 1  |-  ( ( R  o.  R ) 
C_  R  <->  A. x  e.  A  A. y  e.  B  A. z  e.  C  ( (
x R y  /\  y R z )  ->  x R z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wral 2749    i^i cin 3415    C_ wss 3416   class class class wbr 4416   dom cdm 4853   ran crn 4854    o. ccom 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864
This theorem is referenced by:  cotr3  13091
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