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Theorem ertr 7362
Description: An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
ersymb.1  |-  ( ph  ->  R  Er  X )
Assertion
Ref Expression
ertr  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )

Proof of Theorem ertr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ersymb.1 . . . . . . 7  |-  ( ph  ->  R  Er  X )
2 errel 7356 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 17 . . . . . 6  |-  ( ph  ->  Rel  R )
4 simpr 459 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  B R C )
5 brrelex 4861 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
63, 4, 5syl2an 475 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  B  e.  _V )
7 simpr 459 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
8 breq2 4398 . . . . . . 7  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
9 breq1 4397 . . . . . . 7  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
108, 9anbi12d 709 . . . . . 6  |-  ( x  =  B  ->  (
( A R x  /\  x R C )  <->  ( A R B  /\  B R C ) ) )
1110spcegv 3144 . . . . 5  |-  ( B  e.  _V  ->  (
( A R B  /\  B R C )  ->  E. x
( A R x  /\  x R C ) ) )
126, 7, 11sylc 59 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  E. x ( A R x  /\  x R C ) )
13 simpl 455 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  A R B )
14 brrelex 4861 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
153, 13, 14syl2an 475 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A  e.  _V )
16 brrelex2 4862 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
173, 4, 16syl2an 475 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  C  e.  _V )
18 brcog 4989 . . . . 5  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
1915, 17, 18syl2anc 659 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
2012, 19mpbird 232 . . 3  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A ( R  o.  R ) C )
2120ex 432 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A
( R  o.  R
) C ) )
22 df-er 7347 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
2322simp3bi 1014 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
241, 23syl 17 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
2524unssbd 3620 . . 3  |-  ( ph  ->  ( R  o.  R
)  C_  R )
2625ssbrd 4435 . 2  |-  ( ph  ->  ( A ( R  o.  R ) C  ->  A R C ) )
2721, 26syld 42 1  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    u. cun 3411    C_ wss 3413   class class class wbr 4394   `'ccnv 4821   dom cdm 4822    o. ccom 4826   Rel wrel 4827    Er wer 7344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-co 4831  df-er 7347
This theorem is referenced by:  ertrd  7363  erth  7392  iiner  7419  entr  7604  efginvrel2  17067  efgsrel  17074
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