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Theorem erinxp 6937
 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 3522 . . . 4
2 relxp 4942 . . . 4
3 relss 4922 . . . 4
41, 2, 3mp2 9 . . 3
54a1i 11 . 2
6 simpr 448 . . . . 5
7 brinxp2 4898 . . . . 5
86, 7sylib 189 . . . 4
98simp2d 970 . . 3
108simp1d 969 . . 3
11 erinxp.r . . . . 5
1211adantr 452 . . . 4
138simp3d 971 . . . 4
1412, 13ersym 6876 . . 3
15 brinxp2 4898 . . 3
169, 10, 14, 15syl3anbrc 1138 . 2
1710adantrr 698 . . 3
18 simprr 734 . . . . 5
19 brinxp2 4898 . . . . 5
2018, 19sylib 189 . . . 4
2120simp2d 970 . . 3
2211adantr 452 . . . 4
2313adantrr 698 . . . 4
2420simp3d 971 . . . 4
2522, 23, 24ertrd 6880 . . 3
26 brinxp2 4898 . . 3
2717, 21, 25, 26syl3anbrc 1138 . 2
2811adantr 452 . . . . . 6
29 erinxp.a . . . . . . 7
3029sselda 3308 . . . . . 6
3128, 30erref 6884 . . . . 5
3231ex 424 . . . 4
3332pm4.71rd 617 . . 3
34 brin 4219 . . . 4
35 brxp 4868 . . . . . 6
36 anidm 626 . . . . . 6
3735, 36bitri 241 . . . . 5
3837anbi2i 676 . . . 4
3934, 38bitri 241 . . 3
4033, 39syl6bbr 255 . 2
415, 16, 27, 40iserd 6890 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wcel 1721   cin 3279   wss 3280   class class class wbr 4172   cxp 4835   wrel 4842   wer 6861 This theorem is referenced by:  frgpuplem  15359  pi1buni  19018  pi1addf  19025  pi1addval  19026  pi1grplem  19027 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-er 6864
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