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Theorem coepr 35753
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 3484 . . . . . 6 𝑥 ∈ V
31, 2brcnv 5893 . . . . 5 (𝐴 E 𝑥𝑥 E 𝐴)
41epeli 5586 . . . . 5 (𝑥 E 𝐴𝑥𝐴)
53, 4bitri 275 . . . 4 (𝐴 E 𝑥𝑥𝐴)
65anbi1i 624 . . 3 ((𝐴 E 𝑥𝑥𝑅𝐵) ↔ (𝑥𝐴𝑥𝑅𝐵))
76exbii 1848 . 2 (∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝐵))
8 coep.2 . . 3 𝐵 ∈ V
91, 8brco 5881 . 2 (𝐴(𝑅 E )𝐵 ↔ ∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵))
10 df-rex 3071 . 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 206  wa 395  wex 1779  wcel 2108  wrex 3070  Vcvv 3480   class class class wbr 5143   E cep 5583  ccnv 5684  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-cnv 5693  df-co 5694
This theorem is referenced by:  elfuns  35916  brub  35955
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