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Theorem eqbrrdva 5015
 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 4952 . . . 4
31, 2syl6ss 3473 . . 3
4 df-rel 4852 . . 3
53, 4sylibr 215 . 2
6 eqbrrdva.2 . . . 4
76, 2syl6ss 3473 . . 3
8 df-rel 4852 . . 3
97, 8sylibr 215 . 2
101ssbrd 4458 . . . 4
11 brxp 4876 . . . 4
1210, 11syl6ib 229 . . 3
136ssbrd 4458 . . . 4
1413, 11syl6ib 229 . . 3
15 eqbrrdva.3 . . . 4
16153expib 1208 . . 3
1712, 14, 16pm5.21ndd 355 . 2
185, 9, 17eqbrrdv 4943 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   w3a 982   wceq 1437   wcel 1867  cvv 3078   wss 3433   class class class wbr 4417   cxp 4843   wrel 4850 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852 This theorem is referenced by:  metustsym  21494
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