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Theorem eqbrrdva 4704
 Description: Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
Hypotheses
Ref Expression
eqbrrdva.1
eqbrrdva.2
eqbrrdva.3
Assertion
Ref Expression
eqbrrdva
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem eqbrrdva
StepHypRef Expression
1 eqbrrdva.1 . . . 4
2 xpss 4642 . . . 4
31, 2sstrdi 3104 . . 3
4 df-rel 4541 . . 3
53, 4sylibr 133 . 2
6 eqbrrdva.2 . . . 4
76, 2sstrdi 3104 . . 3
8 df-rel 4541 . . 3
97, 8sylibr 133 . 2
101ssbrd 3966 . . . 4
11 brxp 4565 . . . 4
1210, 11syl6ib 160 . . 3
136ssbrd 3966 . . . 4
1413, 11syl6ib 160 . . 3
15 eqbrrdva.3 . . . 4
16153expib 1184 . . 3
1712, 14, 16pm5.21ndd 694 . 2
185, 9, 17eqbrrdv 4631 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   w3a 962   wceq 1331   wcel 1480  cvv 2681   wss 3066   class class class wbr 3924   cxp 4532   wrel 4539 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541 This theorem is referenced by: (None)
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