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Theorem erinxp 6116
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  R  Er
erinxp.a  C_
Assertion
Ref Expression
erinxp  R  i^i  X.  Er

Proof of Theorem erinxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3152 . . . 4  R  i^i  X.  C_  X.
2 relxp 4390 . . . 4  Rel  X.
3 relss 4370 . . . 4  R  i^i  X. 
C_  X.  Rel  X.  Rel  R  i^i  X.
41, 2, 3mp2 16 . . 3  Rel  R  i^i  X.
54a1i 9 . 2  Rel  R  i^i  X.
6 simpr 103 . . . . 5  R  i^i  X.  R  i^i  X.
7 brinxp2 4350 . . . . 5  R  i^i  X.  R
86, 7sylib 127 . . . 4  R  i^i  X.  R
98simp2d 916 . . 3  R  i^i  X.
108simp1d 915 . . 3  R  i^i  X.
11 erinxp.r . . . . 5  R  Er
1211adantr 261 . . . 4  R  i^i  X.  R  Er
138simp3d 917 . . . 4  R  i^i  X.  R
1412, 13ersym 6054 . . 3  R  i^i  X.  R
15 brinxp2 4350 . . 3  R  i^i  X.  R
169, 10, 14, 15syl3anbrc 1087 . 2  R  i^i  X.  R  i^i  X.
1710adantrr 448 . . 3  R  i^i  X.  R  i^i  X.
18 simprr 484 . . . . 5  R  i^i  X.  R  i^i  X.  R  i^i  X.
19 brinxp2 4350 . . . . 5  R  i^i  X.  R
2018, 19sylib 127 . . . 4  R  i^i  X.  R  i^i  X.  R
2120simp2d 916 . . 3  R  i^i  X.  R  i^i  X.
2211adantr 261 . . . 4  R  i^i  X.  R  i^i  X. 
R  Er
2313adantrr 448 . . . 4  R  i^i  X.  R  i^i  X.  R
2420simp3d 917 . . . 4  R  i^i  X.  R  i^i  X.  R
2522, 23, 24ertrd 6058 . . 3  R  i^i  X.  R  i^i  X.  R
26 brinxp2 4350 . . 3  R  i^i  X.  R
2717, 21, 25, 26syl3anbrc 1087 . 2  R  i^i  X.  R  i^i  X.  R  i^i  X.
2811adantr 261 . . . . . 6  R  Er
29 erinxp.a . . . . . . 7  C_
3029sselda 2939 . . . . . 6
3128, 30erref 6062 . . . . 5  R
3231ex 108 . . . 4  R
3332pm4.71rd 374 . . 3  R
34 brin 3802 . . . 4  R  i^i  X.  R  X.
35 brxp 4318 . . . . . 6  X.
36 anidm 376 . . . . . 6
3735, 36bitri 173 . . . . 5  X.
3837anbi2i 430 . . . 4  R  X.  R
3934, 38bitri 173 . . 3  R  i^i  X.  R
4033, 39syl6bbr 187 . 2  R  i^i  X.
415, 16, 27, 40iserd 6068 1  R  i^i  X.  Er
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wcel 1390    i^i cin 2910    C_ wss 2911   class class class wbr 3755    X. cxp 4286   Rel wrel 4293    Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042
This theorem is referenced by: (None)
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