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Theorem ertr 7378
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 7372 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 17 . . . . . 6  |-  ( ph  ->  Rel  R )
4 simpr 463 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  B R C )
5 brrelex 4873 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
63, 4, 5syl2an 480 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  B  e.  _V )
7 simpr 463 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
8 breq2 4406 . . . . . . 7  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
9 breq1 4405 . . . . . . 7  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
108, 9anbi12d 717 . . . . . 6  |-  ( x  =  B  ->  (
( A R x  /\  x R C )  <->  ( A R B  /\  B R C ) ) )
1110spcegv 3135 . . . . 5  |-  ( B  e.  _V  ->  (
( A R B  /\  B R C )  ->  E. x
( A R x  /\  x R C ) ) )
126, 7, 11sylc 62 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  E. x ( A R x  /\  x R C ) )
13 simpl 459 . . . . . 6  |-  ( ( A R B  /\  B R C )  ->  A R B )
14 brrelex 4873 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
153, 13, 14syl2an 480 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A  e.  _V )
16 brrelex2 4874 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
173, 4, 16syl2an 480 . . . . 5  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  C  e.  _V )
18 brcog 5001 . . . . 5  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
1915, 17, 18syl2anc 667 . . . 4  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  -> 
( A ( R  o.  R ) C  <->  E. x ( A R x  /\  x R C ) ) )
2012, 19mpbird 236 . . 3  |-  ( (
ph  /\  ( A R B  /\  B R C ) )  ->  A ( R  o.  R ) C )
2120ex 436 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A
( R  o.  R
) C ) )
22 df-er 7363 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
2322simp3bi 1025 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
241, 23syl 17 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
2524unssbd 3612 . . 3  |-  ( ph  ->  ( R  o.  R
)  C_  R )
2625ssbrd 4444 . 2  |-  ( ph  ->  ( A ( R  o.  R ) C  ->  A R C ) )
2721, 26syld 45 1  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    u. cun 3402    C_ wss 3404   class class class wbr 4402   `'ccnv 4833   dom cdm 4834    o. ccom 4838   Rel wrel 4839    Er wer 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-co 4843  df-er 7363
This theorem is referenced by:  ertrd  7379  erth  7408  iiner  7435  entr  7621  efginvrel2  17377  efgsrel  17384
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