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Theorem coepr 34365
Description: Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
coep.1 𝐴 ∈ V
coep.2 𝐵 ∈ V
Assertion
Ref Expression
coepr (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem coepr
StepHypRef Expression
1 coep.1 . . . . . 6 𝐴 ∈ V
2 vex 3452 . . . . . 6 𝑥 ∈ V
31, 2brcnv 5843 . . . . 5 (𝐴 E 𝑥𝑥 E 𝐴)
41epeli 5544 . . . . 5 (𝑥 E 𝐴𝑥𝐴)
53, 4bitri 275 . . . 4 (𝐴 E 𝑥𝑥𝐴)
65anbi1i 625 . . 3 ((𝐴 E 𝑥𝑥𝑅𝐵) ↔ (𝑥𝐴𝑥𝑅𝐵))
76exbii 1851 . 2 (∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝐵))
8 coep.2 . . 3 𝐵 ∈ V
91, 8brco 5831 . 2 (𝐴(𝑅 E )𝐵 ↔ ∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵))
10 df-rex 3075 . 2 (∃𝑥𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝐵))
117, 9, 103bitr4i 303 1 (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1782  wcel 2107  wrex 3074  Vcvv 3448   class class class wbr 5110   E cep 5541  ccnv 5637  ccom 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-eprel 5542  df-cnv 5646  df-co 5647
This theorem is referenced by:  elfuns  34529  brub  34568
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