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Theorem elima2 5378
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
Hypothesis
Ref Expression
elima.1 𝐴 ∈ V
Assertion
Ref Expression
elima2 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elima2
StepHypRef Expression
1 elima.1 . . 3 𝐴 ∈ V
21elima 5377 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)
3 df-rex 2902 . 2 (∃𝑥𝐶 𝑥𝐵𝐴 ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
42, 3bitri 263 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wex 1695  wcel 1977  wrex 2897  Vcvv 3173   class class class wbr 4578  cima 5031
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 4704  ax-nul 4712  ax-pr 4828
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 4579  df-opab 4639  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041
This theorem is referenced by:  elima3  5379  dminss  5452  imainss  5453  imadif  5873  metcld2  22858  isch2  27258  dfdm5  30715  dfrn5  30716  brimg  31008
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