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Theorem reftr 25796
Description: Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)
Assertion
Ref Expression
reftr  |-  ( ( A Ref B  /\  B Ref C )  ->  A Ref C )

Proof of Theorem reftr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2366 . . . 4  |-  U. A  =  U. A
2 eqid 2366 . . . 4  |-  U. B  =  U. B
31, 2refbas 25787 . . 3  |-  ( A Ref B  ->  U. A  =  U. B )
4 eqid 2366 . . . 4  |-  U. C  =  U. C
52, 4refbas 25787 . . 3  |-  ( B Ref C  ->  U. B  =  U. C )
63, 5sylan9eq 2418 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  U. A  =  U. C )
7 refssex 25788 . . . . . 6  |-  ( ( B Ref C  /\  x  e.  C )  ->  E. y  e.  B  x  C_  y )
87ex 423 . . . . 5  |-  ( B Ref C  ->  (
x  e.  C  ->  E. y  e.  B  x  C_  y ) )
98adantl 452 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. y  e.  B  x  C_  y ) )
10 refssex 25788 . . . . . . . 8  |-  ( ( A Ref B  /\  y  e.  B )  ->  E. z  e.  A  y  C_  z )
1110ad2ant2r 727 . . . . . . 7  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  y  C_  z )
12 sstr2 3272 . . . . . . . . 9  |-  ( x 
C_  y  ->  (
y  C_  z  ->  x 
C_  z ) )
1312reximdv 2739 . . . . . . . 8  |-  ( x 
C_  y  ->  ( E. z  e.  A  y  C_  z  ->  E. z  e.  A  x  C_  z
) )
1413ad2antll 709 . . . . . . 7  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  -> 
( E. z  e.  A  y  C_  z  ->  E. z  e.  A  x  C_  z ) )
1511, 14mpd 14 . . . . . 6  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  x  C_  z )
1615exp32 588 . . . . 5  |-  ( ( A Ref B  /\  B Ref C )  -> 
( y  e.  B  ->  ( x  C_  y  ->  E. z  e.  A  x  C_  z ) ) )
1716rexlimdv 2751 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( E. y  e.  B  x  C_  y  ->  E. z  e.  A  x  C_  z ) )
189, 17syld 40 . . 3  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. z  e.  A  x  C_  z ) )
1918ralrimiv 2710 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  A. x  e.  C  E. z  e.  A  x  C_  z )
20 refrel 25785 . . . . 5  |-  Rel  Ref
2120brrelex2i 4833 . . . 4  |-  ( B Ref C  ->  C  e.  _V )
2221adantl 452 . . 3  |-  ( ( A Ref B  /\  B Ref C )  ->  C  e.  _V )
231, 4isref 25786 . . 3  |-  ( C  e.  _V  ->  ( A Ref C  <->  ( U. A  =  U. C  /\  A. x  e.  C  E. z  e.  A  x  C_  z ) ) )
2422, 23syl 15 . 2  |-  ( ( A Ref B  /\  B Ref C )  -> 
( A Ref C  <->  ( U. A  =  U. C  /\  A. x  e.  C  E. z  e.  A  x  C_  z
) ) )
256, 19, 24mpbir2and 888 1  |-  ( ( A Ref B  /\  B Ref C )  ->  A Ref C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   _Vcvv 2873    C_ wss 3238   U.cuni 3929   class class class wbr 4125   Refcref 25767
This theorem is referenced by:  refssfne  25801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-ref 25771
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