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Theorem elima2 5922
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 5921 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)
3 df-rex 3139 . 2 (∃𝑥𝐶 𝑥𝐵𝐴 ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
42, 3bitri 278 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1781  wcel 2115  wrex 3134  Vcvv 3480   class class class wbr 5052  cima 5545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555
This theorem is referenced by:  elima3  5923  dminss  5997  imainss  5998  imadif  6426  metcld2  23914  isch2  29009  dfdm5  33073  dfrn5  33074  brimg  33455
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