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Theorem coep 35341
Description: Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
coep.1 𝐴 ∈ V
coep.2 𝐵 ∈ V
Assertion
Ref Expression
coep (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem coep
StepHypRef Expression
1 coep.2 . . . . 5 𝐵 ∈ V
21epeli 5579 . . . 4 (𝑥 E 𝐵𝑥𝐵)
32anbi1ci 625 . . 3 ((𝐴𝑅𝑥𝑥 E 𝐵) ↔ (𝑥𝐵𝐴𝑅𝑥))
43exbii 1843 . 2 (∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
5 coep.1 . . 3 𝐴 ∈ V
65, 1brco 5868 . 2 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵))
7 df-rex 3067 . 2 (∃𝑥𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
84, 6, 73bitr4i 303 1 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wex 1774  wcel 2099  wrex 3066  Vcvv 3470   class class class wbr 5143   E cep 5576  ccom 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-eprel 5577  df-co 5682
This theorem is referenced by:  dffr5  35343  brbigcup  35489  elfuns  35506  brimage  35517
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