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Theorem erinxp 6733
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 3390 . . . 4  |-  ( R  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2 relxp 4794 . . . 4  |-  Rel  ( B  X.  B )
3 relss 4775 . . . 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 17 . . 3  |-  Rel  ( R  i^i  ( B  X.  B ) )
54a1i 10 . 2  |-  ( ph  ->  Rel  ( R  i^i  ( B  X.  B
) ) )
6 simpr 447 . . . . 5  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x ( R  i^i  ( B  X.  B ) ) y )
7 brinxp2 4751 . . . . 5  |-  ( x ( R  i^i  ( B  X.  B ) ) y  <->  ( x  e.  B  /\  y  e.  B  /\  x R y ) )
86, 7sylib 188 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  ( x  e.  B  /\  y  e.  B  /\  x R y ) )
98simp2d 968 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y  e.  B )
108simp1d 967 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x  e.  B )
11 erinxp.r . . . . 5  |-  ( ph  ->  R  Er  A )
1211adantr 451 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  R  Er  A )
138simp3d 969 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x R
y )
1412, 13ersym 6672 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y R x )
15 brinxp2 4751 . . 3  |-  ( y ( R  i^i  ( B  X.  B ) ) x  <->  ( y  e.  B  /\  x  e.  B  /\  y R x ) )
169, 10, 14, 15syl3anbrc 1136 . 2  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y ( R  i^i  ( B  X.  B ) ) x )
1710adantrr 697 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x  e.  B )
18 simprr 733 . . . . 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 4751 . . . . 5  |-  ( y ( R  i^i  ( B  X.  B ) ) z  <->  ( y  e.  B  /\  z  e.  B  /\  y R z ) )
2018, 19sylib 188 . . . 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 968 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
z  e.  B )
2211adantr 451 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  R  Er  A )
2313adantrr 697 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x R y )
2420simp3d 969 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
y R z )
2522, 23, 24ertrd 6676 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x R z )
26 brinxp2 4751 . . 3  |-  ( x ( R  i^i  ( B  X.  B ) ) z  <->  ( x  e.  B  /\  z  e.  B  /\  x R z ) )
2717, 21, 25, 26syl3anbrc 1136 . 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 451 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  R  Er  A )
29 erinxp.a . . . . . . 7  |-  ( ph  ->  B  C_  A )
3029sselda 3180 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
3128, 30erref 6680 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  x R x )
3231ex 423 . . . 4  |-  ( ph  ->  ( x  e.  B  ->  x R x ) )
3332pm4.71rd 616 . . 3  |-  ( ph  ->  ( x  e.  B  <->  ( x R x  /\  x  e.  B )
) )
34 brin 4070 . . . 4  |-  ( x ( R  i^i  ( B  X.  B ) ) x  <->  ( x R x  /\  x ( B  X.  B ) x ) )
35 brxp 4720 . . . . . 6  |-  ( x ( B  X.  B
) x  <->  ( x  e.  B  /\  x  e.  B ) )
36 anidm 625 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  B )  <->  x  e.  B )
3735, 36bitri 240 . . . . 5  |-  ( x ( B  X.  B
) x  <->  x  e.  B )
3837anbi2i 675 . . . 4  |-  ( ( x R x  /\  x ( B  X.  B ) x )  <-> 
( x R x  /\  x  e.  B
) )
3934, 38bitri 240 . . 3  |-  ( x ( R  i^i  ( B  X.  B ) ) x  <->  ( x R x  /\  x  e.  B ) )
4033, 39syl6bbr 254 . 2  |-  ( ph  ->  ( x  e.  B  <->  x ( R  i^i  ( B  X.  B ) ) x ) )
415, 16, 27, 40iserd 6686 1  |-  ( ph  ->  ( R  i^i  ( B  X.  B ) )  Er  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023    X. cxp 4687   Rel wrel 4694    Er wer 6657
This theorem is referenced by:  frgpuplem  15081  pi1buni  18538  pi1addf  18545  pi1addval  18546  pi1grplem  18547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-er 6660
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