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Theorem ertr 6287
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 6281 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 14 . . . . . 6  |-  ( ph  ->  Rel  R )
4 simpr 108 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  B R C )
5 brrelex 4466 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
63, 4, 5syl2an 283 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  B  e.  _V )
7 simpr 108 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
8 breq2 3841 . . . . . . 7  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
9 breq1 3840 . . . . . . 7  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
108, 9anbi12d 457 . . . . . 6  |-  ( x  =  B  ->  (
( A R x  /\  x R C )  <->  ( A R B  /\  B R C ) ) )
1110spcegv 2707 . . . . 5  |-  ( B  e.  _V  ->  (
( A R B  /\  B R C )  ->  E. x
( A R x  /\  x R C ) ) )
126, 7, 11sylc 61 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  E. x ( A R x  /\  x R C ) )
13 simpl 107 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  A R B )
14 brrelex 4466 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
153, 13, 14syl2an 283 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A  e.  _V )
16 brrelex2 4467 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
173, 4, 16syl2an 283 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  C  e.  _V )
18 brcog 4591 . . . . 5  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
1915, 17, 18syl2anc 403 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
2012, 19mpbird 165 . . 3  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A ( R  o.  R ) C )
2120ex 113 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A
( R  o.  R
) C ) )
22 df-er 6272 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
2322simp3bi 960 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
241, 23syl 14 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
2524unssbd 3176 . . 3  |-  ( ph  ->  ( R  o.  R
)  C_  R )
2625ssbrd 3878 . 2  |-  ( ph  ->  ( A ( R  o.  R ) C  ->  A R C ) )
2721, 26syld 44 1  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619    u. cun 2995    C_ wss 2997   class class class wbr 3837   `'ccnv 4427   dom cdm 4428    o. ccom 4432   Rel wrel 4433    Er wer 6269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-co 4437  df-er 6272
This theorem is referenced by:  ertrd  6288  erth  6316  iinerm  6344  entr  6481
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