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Theorem reftr 26383
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 2438 . . . 4  |-  U. A  =  U. A
2 eqid 2438 . . . 4  |-  U. B  =  U. B
31, 2refbas 26374 . . 3  |-  ( A Ref B  ->  U. A  =  U. B )
4 eqid 2438 . . . 4  |-  U. C  =  U. C
52, 4refbas 26374 . . 3  |-  ( B Ref C  ->  U. B  =  U. C )
63, 5sylan9eq 2490 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  U. A  =  U. C )
7 refssex 26375 . . . . . 6  |-  ( ( B Ref C  /\  x  e.  C )  ->  E. y  e.  B  x  C_  y )
87ex 425 . . . . 5  |-  ( B Ref C  ->  (
x  e.  C  ->  E. y  e.  B  x  C_  y ) )
98adantl 454 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. y  e.  B  x  C_  y ) )
10 refssex 26375 . . . . . . 7  |-  ( ( A Ref B  /\  y  e.  B )  ->  E. z  e.  A  y  C_  z )
1110ad2ant2r 729 . . . . . 6  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  y  C_  z )
12 sstr2 3357 . . . . . . . 8  |-  ( x 
C_  y  ->  (
y  C_  z  ->  x 
C_  z ) )
1312reximdv 2819 . . . . . . 7  |-  ( x 
C_  y  ->  ( E. z  e.  A  y  C_  z  ->  E. z  e.  A  x  C_  z
) )
1413ad2antll 711 . . . . . 6  |-  ( ( ( 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 15 . . . . 5  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  x  C_  z )
1615rexlimdvaa 2833 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( E. y  e.  B  x  C_  y  ->  E. z  e.  A  x  C_  z ) )
179, 16syld 43 . . 3  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. z  e.  A  x  C_  z ) )
1817ralrimiv 2790 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  A. x  e.  C  E. z  e.  A  x  C_  z )
19 refrel 26372 . . . . 5  |-  Rel  Ref
2019brrelex2i 4922 . . . 4  |-  ( B Ref C  ->  C  e.  _V )
2120adantl 454 . . 3  |-  ( ( A Ref B  /\  B Ref C )  ->  C  e.  _V )
221, 4isref 26373 . . 3  |-  ( C  e.  _V  ->  ( A Ref C  <->  ( U. A  =  U. C  /\  A. x  e.  C  E. z  e.  A  x  C_  z ) ) )
2321, 22syl 16 . 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
) ) )
246, 18, 23mpbir2and 890 1  |-  ( ( A Ref B  /\  B Ref C )  ->  A Ref C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   U.cuni 4017   class class class wbr 4215   Refcref 26354
This theorem is referenced by:  refssfne  26388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-ref 26358
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