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Theorem elima 5627
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 5626 . 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 196  wcel 2137  wrex 3049  Vcvv 3338   class class class wbr 4802  cima 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-xp 5270  df-cnv 5272  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277
This theorem is referenced by:  elima2  5628  rninxp  5729  imaco  5799  isarep1  6136  eliman0  6382  funimass4  6407  isomin  6748  dfsup2  8513  dfac10b  9151  hausmapdom  21503  pi1blem  23037  adjbd1o  29251  elintfv  31967  imaindm  31985  scutun12  32221  madeval2  32240  brimage  32337  dfrecs2  32361  dfrdg4  32362  dfint3  32363  imagesset  32364  elimaint  38440  elintima  38445
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