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Theorem erinxp 6937
Description: A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
erinxp.r  |-  ( ph  ->  R  Er  A )
erinxp.a  |-  ( ph  ->  B  C_  A )
Assertion
Ref Expression
erinxp  |-  ( ph  ->  ( R  i^i  ( B  X.  B ) )  Er  B )

Proof of Theorem erinxp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3522 . . . 4  |-  ( R  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2 relxp 4942 . . . 4  |-  Rel  ( B  X.  B )
3 relss 4922 . . . 4  |-  ( ( R  i^i  ( B  X.  B ) ) 
C_  ( B  X.  B )  ->  ( Rel  ( B  X.  B
)  ->  Rel  ( R  i^i  ( B  X.  B ) ) ) )
41, 2, 3mp2 9 . . 3  |-  Rel  ( R  i^i  ( B  X.  B ) )
54a1i 11 . 2  |-  ( ph  ->  Rel  ( R  i^i  ( B  X.  B
) ) )
6 simpr 448 . . . . 5  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x ( R  i^i  ( B  X.  B ) ) y )
7 brinxp2 4898 . . . . 5  |-  ( x ( R  i^i  ( B  X.  B ) ) y  <->  ( x  e.  B  /\  y  e.  B  /\  x R y ) )
86, 7sylib 189 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  ( x  e.  B  /\  y  e.  B  /\  x R y ) )
98simp2d 970 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y  e.  B )
108simp1d 969 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x  e.  B )
11 erinxp.r . . . . 5  |-  ( ph  ->  R  Er  A )
1211adantr 452 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  R  Er  A )
138simp3d 971 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x R
y )
1412, 13ersym 6876 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y R x )
15 brinxp2 4898 . . 3  |-  ( y ( R  i^i  ( B  X.  B ) ) x  <->  ( y  e.  B  /\  x  e.  B  /\  y R x ) )
169, 10, 14, 15syl3anbrc 1138 . 2  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y ( R  i^i  ( B  X.  B ) ) x )
1710adantrr 698 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x  e.  B )
18 simprr 734 . . . . 5  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
y ( R  i^i  ( B  X.  B
) ) z )
19 brinxp2 4898 . . . . 5  |-  ( y ( R  i^i  ( B  X.  B ) ) z  <->  ( y  e.  B  /\  z  e.  B  /\  y R z ) )
2018, 19sylib 189 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
( y  e.  B  /\  z  e.  B  /\  y R z ) )
2120simp2d 970 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
z  e.  B )
2211adantr 452 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  R  Er  A )
2313adantrr 698 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x R y )
2420simp3d 971 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
y R z )
2522, 23, 24ertrd 6880 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x R z )
26 brinxp2 4898 . . 3  |-  ( x ( R  i^i  ( B  X.  B ) ) z  <->  ( x  e.  B  /\  z  e.  B  /\  x R z ) )
2717, 21, 25, 26syl3anbrc 1138 . 2  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x ( R  i^i  ( B  X.  B
) ) z )
2811adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  R  Er  A )
29 erinxp.a . . . . . . 7  |-  ( ph  ->  B  C_  A )
3029sselda 3308 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
3128, 30erref 6884 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  x R x )
3231ex 424 . . . 4  |-  ( ph  ->  ( x  e.  B  ->  x R x ) )
3332pm4.71rd 617 . . 3  |-  ( ph  ->  ( x  e.  B  <->  ( x R x  /\  x  e.  B )
) )
34 brin 4219 . . . 4  |-  ( x ( R  i^i  ( B  X.  B ) ) x  <->  ( x R x  /\  x ( B  X.  B ) x ) )
35 brxp 4868 . . . . . 6  |-  ( x ( B  X.  B
) x  <->  ( x  e.  B  /\  x  e.  B ) )
36 anidm 626 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  B )  <->  x  e.  B )
3735, 36bitri 241 . . . . 5  |-  ( x ( B  X.  B
) x  <->  x  e.  B )
3837anbi2i 676 . . . 4  |-  ( ( x R x  /\  x ( B  X.  B ) x )  <-> 
( x R x  /\  x  e.  B
) )
3934, 38bitri 241 . . 3  |-  ( x ( R  i^i  ( B  X.  B ) ) x  <->  ( x R x  /\  x  e.  B ) )
4033, 39syl6bbr 255 . 2  |-  ( ph  ->  ( x  e.  B  <->  x ( R  i^i  ( B  X.  B ) ) x ) )
415, 16, 27, 40iserd 6890 1  |-  ( ph  ->  ( R  i^i  ( B  X.  B ) )  Er  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721    i^i cin 3279    C_ wss 3280   class class class wbr 4172    X. cxp 4835   Rel wrel 4842    Er wer 6861
This theorem is referenced by:  frgpuplem  15359  pi1buni  19018  pi1addf  19025  pi1addval  19026  pi1grplem  19027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-er 6864
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