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Theorem elima 5688
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 5687 . 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 197  wcel 2157  wrex 3104  Vcvv 3398   class class class wbr 4851  cima 5321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-xp 5324  df-cnv 5326  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331
This theorem is referenced by:  elima2  5689  rninxp  5791  imaco  5861  isarep1  6191  eliman0  6446  funimass4  6471  isomin  6814  dfsup2  8592  dfac10b  9249  hausmapdom  21521  pi1blem  23055  adjbd1o  29278  elintfv  31989  imaindm  32007  scutun12  32243  madeval2  32262  brimage  32359  dfrecs2  32383  dfrdg4  32384  dfint3  32385  imagesset  32386  elimaint  38441  elintima  38446
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