ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  erinxp Unicode version

Theorem erinxp 6554
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 3328 . . . 4  |-  ( R  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2 relxp 4695 . . . 4  |-  Rel  ( B  X.  B )
3 relss 4673 . . . 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 16 . . 3  |-  Rel  ( R  i^i  ( B  X.  B ) )
54a1i 9 . 2  |-  ( ph  ->  Rel  ( R  i^i  ( B  X.  B
) ) )
6 simpr 109 . . . . 5  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x ( R  i^i  ( B  X.  B ) ) y )
7 brinxp2 4653 . . . . 5  |-  ( x ( R  i^i  ( B  X.  B ) ) y  <->  ( x  e.  B  /\  y  e.  B  /\  x R y ) )
86, 7sylib 121 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  ( x  e.  B  /\  y  e.  B  /\  x R y ) )
98simp2d 995 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y  e.  B )
108simp1d 994 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x  e.  B )
11 erinxp.r . . . . 5  |-  ( ph  ->  R  Er  A )
1211adantr 274 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  R  Er  A )
138simp3d 996 . . . 4  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  x R
y )
1412, 13ersym 6492 . . 3  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y R x )
15 brinxp2 4653 . . 3  |-  ( y ( R  i^i  ( B  X.  B ) ) x  <->  ( y  e.  B  /\  x  e.  B  /\  y R x ) )
169, 10, 14, 15syl3anbrc 1166 . 2  |-  ( (
ph  /\  x ( R  i^i  ( B  X.  B ) ) y )  ->  y ( R  i^i  ( B  X.  B ) ) x )
1710adantrr 471 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x  e.  B )
18 simprr 522 . . . . 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 4653 . . . . 5  |-  ( y ( R  i^i  ( B  X.  B ) ) z  <->  ( y  e.  B  /\  z  e.  B  /\  y R z ) )
2018, 19sylib 121 . . . 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 995 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
z  e.  B )
2211adantr 274 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  R  Er  A )
2313adantrr 471 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x R y )
2420simp3d 996 . . . 4  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  -> 
y R z )
2522, 23, 24ertrd 6496 . . 3  |-  ( (
ph  /\  ( x
( R  i^i  ( B  X.  B ) ) y  /\  y ( R  i^i  ( B  X.  B ) ) z ) )  ->  x R z )
26 brinxp2 4653 . . 3  |-  ( x ( R  i^i  ( B  X.  B ) ) z  <->  ( x  e.  B  /\  z  e.  B  /\  x R z ) )
2717, 21, 25, 26syl3anbrc 1166 . 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 274 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  R  Er  A )
29 erinxp.a . . . . . . 7  |-  ( ph  ->  B  C_  A )
3029sselda 3128 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
3128, 30erref 6500 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  x R x )
3231ex 114 . . . 4  |-  ( ph  ->  ( x  e.  B  ->  x R x ) )
3332pm4.71rd 392 . . 3  |-  ( ph  ->  ( x  e.  B  <->  ( x R x  /\  x  e.  B )
) )
34 brin 4016 . . . 4  |-  ( x ( R  i^i  ( B  X.  B ) ) x  <->  ( x R x  /\  x ( B  X.  B ) x ) )
35 brxp 4617 . . . . . 6  |-  ( x ( B  X.  B
) x  <->  ( x  e.  B  /\  x  e.  B ) )
36 anidm 394 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  B )  <->  x  e.  B )
3735, 36bitri 183 . . . . 5  |-  ( x ( B  X.  B
) x  <->  x  e.  B )
3837anbi2i 453 . . . 4  |-  ( ( x R x  /\  x ( B  X.  B ) x )  <-> 
( x R x  /\  x  e.  B
) )
3934, 38bitri 183 . . 3  |-  ( x ( R  i^i  ( B  X.  B ) ) x  <->  ( x R x  /\  x  e.  B ) )
4033, 39bitr4di 197 . 2  |-  ( ph  ->  ( x  e.  B  <->  x ( R  i^i  ( B  X.  B ) ) x ) )
415, 16, 27, 40iserd 6506 1  |-  ( ph  ->  ( R  i^i  ( B  X.  B ) )  Er  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    e. wcel 2128    i^i cin 3101    C_ wss 3102   class class class wbr 3965    X. cxp 4584   Rel wrel 4591    Er wer 6477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-er 6480
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator