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Theorem cotr3 13117
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  =  ( A  i^i  C
)
cotr3.c  |-  C  =  ran  R
Assertion
Ref Expression
cotr3  |-  ( ( 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 cotr3
StepHypRef Expression
1 cotr3.a . . 3  |-  A  =  dom  R
21eqimss2i 3473 . 2  |-  dom  R  C_  A
3 cotr3.b . . . 4  |-  B  =  ( A  i^i  C
)
4 cotr3.c . . . . 5  |-  C  =  ran  R
51, 4ineq12i 3623 . . . 4  |-  ( A  i^i  C )  =  ( dom  R  i^i  ran 
R )
63, 5eqtri 2493 . . 3  |-  B  =  ( dom  R  i^i  ran 
R )
76eqimss2i 3473 . 2  |-  ( dom 
R  i^i  ran  R ) 
C_  B
84eqimss2i 3473 . 2  |-  ran  R  C_  C
92, 7, 8cotr2 13116 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 376    = wceq 1452   A.wral 2756    i^i cin 3389    C_ wss 3390   class class class wbr 4395   dom cdm 4839   ran crn 4840    o. ccom 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850
This theorem is referenced by: (None)
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