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Theorem erinxp 6978
 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 3562 . . . 4
2 relxp 4983 . . . 4
3 relss 4963 . . . 4
41, 2, 3mp2 9 . . 3
54a1i 11 . 2
6 simpr 448 . . . . 5
7 brinxp2 4939 . . . . 5
86, 7sylib 189 . . . 4
98simp2d 970 . . 3
108simp1d 969 . . 3
11 erinxp.r . . . . 5
1211adantr 452 . . . 4
138simp3d 971 . . . 4
1412, 13ersym 6917 . . 3
15 brinxp2 4939 . . 3
169, 10, 14, 15syl3anbrc 1138 . 2
18 simprr 734 . . . . 5
19 brinxp2 4939 . . . . 5
2018, 19sylib 189 . . . 4
2120simp2d 970 . . 3
2211adantr 452 . . . 4
2313adantrr 698 . . . 4
2420simp3d 971 . . . 4
2522, 23, 24ertrd 6921 . . 3
26 brinxp2 4939 . . 3
2717, 21, 25, 26syl3anbrc 1138 . 2
2811adantr 452 . . . . . 6
29 erinxp.a . . . . . . 7
3029sselda 3348 . . . . . 6
3128, 30erref 6925 . . . . 5
3231ex 424 . . . 4
3332pm4.71rd 617 . . 3
34 brin 4259 . . . 4
35 brxp 4909 . . . . . 6
36 anidm 626 . . . . . 6
3735, 36bitri 241 . . . . 5
3837anbi2i 676 . . . 4
3934, 38bitri 241 . . 3
4033, 39syl6bbr 255 . 2
415, 16, 27, 40iserd 6931 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wcel 1725   cin 3319   wss 3320   class class class wbr 4212   cxp 4876   wrel 4883   wer 6902 This theorem is referenced by:  frgpuplem  15404  pi1buni  19065  pi1addf  19072  pi1addval  19073  pi1grplem  19074 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-er 6905
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