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Theorem elimag 6015
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
Assertion
Ref Expression
elimag (𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elimag
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq2 5107 . . 3 (𝑦 = 𝐴 → (𝑥𝐵𝑦𝑥𝐵𝐴))
21rexbidv 3173 . 2 (𝑦 = 𝐴 → (∃𝑥𝐶 𝑥𝐵𝑦 ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
3 dfima2 6013 . 2 (𝐵𝐶) = {𝑦 ∣ ∃𝑥𝐶 𝑥𝐵𝑦}
42, 3elab2g 3630 1 (𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wrex 3071   class class class wbr 5103  cima 5634
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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644
This theorem is referenced by:  elima  6016  fvelima  6905  fvelimad  6906  opelco3  34165  fvelima2  43393  afvelima  45300
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