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Theorem reftr 20123
Description: Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
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 2396 . . . 4  |-  U. B  =  U. B
2 eqid 2396 . . . 4  |-  U. C  =  U. C
31, 2refbas 20119 . . 3  |-  ( B Ref C  ->  U. C  =  U. B )
4 eqid 2396 . . . 4  |-  U. A  =  U. A
54, 1refbas 20119 . . 3  |-  ( A Ref B  ->  U. B  =  U. A )
63, 5sylan9eqr 2459 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  U. C  =  U. A )
7 refssex 20120 . . . . . 6  |-  ( ( A Ref B  /\  x  e.  A )  ->  E. y  e.  B  x  C_  y )
87ex 432 . . . . 5  |-  ( A Ref B  ->  (
x  e.  A  ->  E. y  e.  B  x  C_  y ) )
98adantr 463 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  A  ->  E. y  e.  B  x  C_  y ) )
10 refssex 20120 . . . . . . 7  |-  ( ( B Ref C  /\  y  e.  B )  ->  E. z  e.  C  y  C_  z )
1110ad2ant2lr 745 . . . . . 6  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  C  y  C_  z )
12 sstr2 3441 . . . . . . . 8  |-  ( x 
C_  y  ->  (
y  C_  z  ->  x 
C_  z ) )
1312reximdv 2870 . . . . . . 7  |-  ( x 
C_  y  ->  ( E. z  e.  C  y  C_  z  ->  E. z  e.  C  x  C_  z
) )
1413ad2antll 726 . . . . . 6  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  -> 
( E. z  e.  C  y  C_  z  ->  E. z  e.  C  x  C_  z ) )
1511, 14mpd 15 . . . . 5  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  C  x  C_  z )
1615rexlimdvaa 2889 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( E. y  e.  B  x  C_  y  ->  E. z  e.  C  x  C_  z ) )
179, 16syld 44 . . 3  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  A  ->  E. z  e.  C  x  C_  z ) )
1817ralrimiv 2808 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  A. x  e.  A  E. z  e.  C  x  C_  z )
19 refrel 20117 . . . . 5  |-  Rel  Ref
2019brrelexi 4971 . . . 4  |-  ( A Ref B  ->  A  e.  _V )
2120adantr 463 . . 3  |-  ( ( A Ref B  /\  B Ref C )  ->  A  e.  _V )
224, 2isref 20118 . . 3  |-  ( A  e.  _V  ->  ( A Ref C  <->  ( U. C  =  U. A  /\  A. x  e.  A  E. z  e.  C  x  C_  z ) ) )
2321, 22syl 16 . 2  |-  ( ( A Ref B  /\  B Ref C )  -> 
( A Ref C  <->  ( U. C  =  U. A  /\  A. x  e.  A  E. z  e.  C  x  C_  z
) ) )
246, 18, 23mpbir2and 920 1  |-  ( ( A Ref B  /\  B Ref C )  ->  A Ref C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746   E.wrex 2747   _Vcvv 3051    C_ wss 3406   U.cuni 4180   class class class wbr 4384   Refcref 20111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-xp 4936  df-rel 4937  df-ref 20114
This theorem is referenced by:  refssfne  30382
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