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Theorem elima3 5392
 Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
elima.1 𝐴 ∈ V
Assertion
Ref Expression
elima3 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elima3
StepHypRef Expression
1 elima.1 . . 3 𝐴 ∈ V
21elima2 5391 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
3 df-br 4584 . . . 4 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
43anbi2i 726 . . 3 ((𝑥𝐶𝑥𝐵𝐴) ↔ (𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
54exbii 1764 . 2 (∃𝑥(𝑥𝐶𝑥𝐵𝐴) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
62, 5bitri 263 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  cnvresima  5541  imaiun  6407  1stpreimas  28866  elima4  30924  imaiun1  36962  snhesn  37100
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