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Theorem ertr 6920
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 6914 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  Rel  R )
4 simpr 448 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  B R C )
5 brrelex 4916 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
63, 4, 5syl2an 464 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  B  e.  _V )
7 simpr 448 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
8 breq2 4216 . . . . . . 7  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
9 breq1 4215 . . . . . . 7  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
108, 9anbi12d 692 . . . . . 6  |-  ( x  =  B  ->  (
( A R x  /\  x R C )  <->  ( A R B  /\  B R C ) ) )
1110spcegv 3037 . . . . 5  |-  ( B  e.  _V  ->  (
( A R B  /\  B R C )  ->  E. x
( A R x  /\  x R C ) ) )
126, 7, 11sylc 58 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  E. x ( A R x  /\  x R C ) )
13 simpl 444 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  A R B )
14 brrelex 4916 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
153, 13, 14syl2an 464 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A  e.  _V )
16 brrelex2 4917 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
173, 4, 16syl2an 464 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  C  e.  _V )
18 brcog 5039 . . . . 5  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
1915, 17, 18syl2anc 643 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
2012, 19mpbird 224 . . 3  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A ( R  o.  R ) C )
2120ex 424 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A
( R  o.  R
) C ) )
22 df-er 6905 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
2322simp3bi 974 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
241, 23syl 16 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
2524unssbd 3525 . . 3  |-  ( ph  ->  ( R  o.  R
)  C_  R )
2625ssbrd 4253 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318    C_ wss 3320   class class class wbr 4212   `'ccnv 4877   dom cdm 4878    o. ccom 4882   Rel wrel 4883    Er wer 6902
This theorem is referenced by:  ertrd  6921  erth  6949  iiner  6976  entr  7159  efginvrel2  15359  efgsrel  15366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-co 4887  df-er 6905
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