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Theorem coep 32183
 Description: Composition with epsilon. (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 . . . . . 6 𝐵 ∈ V
21epeli 5257 . . . . 5 (𝑥 E 𝐵𝑥𝐵)
32anbi2i 618 . . . 4 ((𝐴𝑅𝑥𝑥 E 𝐵) ↔ (𝐴𝑅𝑥𝑥𝐵))
4 ancom 454 . . . 4 ((𝐴𝑅𝑥𝑥𝐵) ↔ (𝑥𝐵𝐴𝑅𝑥))
53, 4bitri 267 . . 3 ((𝐴𝑅𝑥𝑥 E 𝐵) ↔ (𝑥𝐵𝐴𝑅𝑥))
65exbii 1949 . 2 (∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
7 coep.1 . . 3 𝐴 ∈ V
87, 1brco 5525 . 2 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵))
9 df-rex 3123 . 2 (∃𝑥𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
106, 8, 93bitr4i 295 1 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386  ∃wex 1880   ∈ wcel 2166  ∃wrex 3118  Vcvv 3414   class class class wbr 4873   E cep 5254   ∘ ccom 5346 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-eprel 5255  df-co 5351 This theorem is referenced by:  dffr5  32185  brbigcup  32544  elfuns  32561  brimage  32572
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