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Theorem elima3 6066
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 6065 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
3 df-br 5149 . . . 4 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
43anbi2i 623 . . 3 ((𝑥𝐶𝑥𝐵𝐴) ↔ (𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
54exbii 1850 . 2 (∃𝑥(𝑥𝐶𝑥𝐵𝐴) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
62, 5bitri 274 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wex 1781  wcel 2106  Vcvv 3474  cop 4634   class class class wbr 5148  cima 5679
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  cnvresima  6229  imaiun  7243  1stpreimas  31922  elima4  34742  imaiun1  42392  snhesn  42527
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