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Theorem eqbrrdva 4448
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  C_  C  X.  D
eqbrrdva.2  C_  C  X.  D
eqbrrdva.3  C  D
Assertion
Ref Expression
eqbrrdva
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:    C(,)    D(,)

Proof of Theorem eqbrrdva
StepHypRef Expression
1 eqbrrdva.1 . . . 4  C_  C  X.  D
2 xpss 4389 . . . 4  C  X.  D  C_  _V  X.  _V
31, 2syl6ss 2951 . . 3  C_  _V  X.  _V
4 df-rel 4295 . . 3  Rel  C_  _V  X.  _V
53, 4sylibr 137 . 2  Rel
6 eqbrrdva.2 . . . 4  C_  C  X.  D
76, 2syl6ss 2951 . . 3  C_  _V  X.  _V
8 df-rel 4295 . . 3  Rel  C_  _V  X.  _V
97, 8sylibr 137 . 2  Rel
101ssbrd 3796 . . . 4  C  X.  D
11 brxp 4318 . . . 4  C  X.  D  C  D
1210, 11syl6ib 150 . . 3  C  D
136ssbrd 3796 . . . 4  C  X.  D
1413, 11syl6ib 150 . . 3  C  D
15 eqbrrdva.3 . . . 4  C  D
16153expib 1106 . . 3  C  D
1712, 14, 16pm5.21ndd 620 . 2
185, 9, 17eqbrrdv 4380 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390   _Vcvv 2551    C_ wss 2911   class class class wbr 3755    X. cxp 4286   Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by: (None)
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