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Theorem erinxp 7404
 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
erinxp.a
Assertion
Ref Expression
erinxp

Proof of Theorem erinxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3718 . . . 4
2 relxp 5115 . . . 4
3 relss 5095 . . . 4
41, 2, 3mp2 9 . . 3
54a1i 11 . 2
6 simpr 461 . . . . 5
7 brinxp2 5066 . . . . 5
86, 7sylib 196 . . . 4
98simp2d 1009 . . 3
108simp1d 1008 . . 3
11 erinxp.r . . . . 5
1211adantr 465 . . . 4
138simp3d 1010 . . . 4
1412, 13ersym 7342 . . 3
15 brinxp2 5066 . . 3
169, 10, 14, 15syl3anbrc 1180 . 2
18 simprr 757 . . . . 5
19 brinxp2 5066 . . . . 5
2018, 19sylib 196 . . . 4
2120simp2d 1009 . . 3
2211adantr 465 . . . 4
2313adantrr 716 . . . 4
2420simp3d 1010 . . . 4
2522, 23, 24ertrd 7346 . . 3
26 brinxp2 5066 . . 3
2717, 21, 25, 26syl3anbrc 1180 . 2
2811adantr 465 . . . . . 6
29 erinxp.a . . . . . . 7
3029sselda 3503 . . . . . 6
3128, 30erref 7350 . . . . 5
3231ex 434 . . . 4
3332pm4.71rd 635 . . 3
34 brin 4501 . . . 4
35 brxp 5035 . . . . . 6
36 anidm 644 . . . . . 6
3735, 36bitri 249 . . . . 5
3837anbi2i 694 . . . 4
3934, 38bitri 249 . . 3
4033, 39syl6bbr 263 . 2
415, 16, 27, 40iserd 7356 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  e.wcel 1818  i^icin 3474  C_wss 3475   class class class wbr 4452  X.cxp 5002  Relwrel 5009  Erwer 7327 This theorem is referenced by:  frgpuplem  16790  pi1buni  21540  pi1addf  21547  pi1addval  21548  pi1grplem  21549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-er 7330
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