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Theorem ertr 6691
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 6685 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 15 . . . . . 6  |-  ( ph  ->  Rel  R )
4 simpr 447 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  B R C )
5 brrelex 4743 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
63, 4, 5syl2an 463 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  B  e.  _V )
7 simpr 447 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
8 breq2 4043 . . . . . . 7  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
9 breq1 4042 . . . . . . 7  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
108, 9anbi12d 691 . . . . . 6  |-  ( x  =  B  ->  (
( A R x  /\  x R C )  <->  ( A R B  /\  B R C ) ) )
1110spcegv 2882 . . . . 5  |-  ( B  e.  _V  ->  (
( A R B  /\  B R C )  ->  E. x
( A R x  /\  x R C ) ) )
126, 7, 11sylc 56 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  E. x ( A R x  /\  x R C ) )
13 simpl 443 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  A R B )
14 brrelex 4743 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
153, 13, 14syl2an 463 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A  e.  _V )
16 brrelex2 4744 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
173, 4, 16syl2an 463 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  C  e.  _V )
18 brcog 4866 . . . . 5  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
1915, 17, 18syl2anc 642 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
2012, 19mpbird 223 . . 3  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A ( R  o.  R ) C )
2120ex 423 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A
( R  o.  R
) C ) )
22 ssun2 3352 . . . 4  |-  ( R  o.  R )  C_  ( `' R  u.  ( R  o.  R )
)
23 df-er 6676 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
2423simp3bi 972 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
251, 24syl 15 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
2622, 25syl5ss 3203 . . 3  |-  ( ph  ->  ( R  o.  R
)  C_  R )
2726ssbrd 4080 . 2  |-  ( ph  ->  ( A ( R  o.  R ) C  ->  A R C ) )
2821, 27syld 40 1  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    C_ wss 3165   class class class wbr 4039   `'ccnv 4704   dom cdm 4705    o. ccom 4709   Rel wrel 4710    Er wer 6673
This theorem is referenced by:  ertrd  6692  erth  6720  iiner  6747  entr  6929  efginvrel2  15052  efgsrel  15059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-co 4714  df-er 6676
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