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Theorem elima 6065
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
Hypothesis
Ref Expression
elima.1 𝐴 ∈ V
Assertion
Ref Expression
elima (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elima
StepHypRef Expression
1 elima.1 . 2 𝐴 ∈ V
2 elimag 6064 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
31, 2ax-mp 5 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2104  wrex 3068  Vcvv 3472   class class class wbr 5149  cima 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  elima2  6066  rninxp  6179  imaco  6251  imaindm  6299  isarep1  6638  isarep1OLD  6639  eliman0  6932  funimass4  6957  isomin  7338  dfsup2  9443  dfac10b  10138  hausmapdom  23226  pi1blem  24788  scutun12  27546  madeval2  27583  adjbd1o  31603  brimage  35200  dfrecs2  35224  dfrdg4  35225  dfint3  35226  imagesset  35227  elimaint  42704  elintima  42708
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