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Theorem elimag 5926
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 5061 . . 3 (𝑦 = 𝐴 → (𝑥𝐵𝑦𝑥𝐵𝐴))
21rexbidv 3294 . 2 (𝑦 = 𝐴 → (∃𝑥𝐶 𝑥𝐵𝑦 ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
3 dfima2 5924 . 2 (𝐵𝐶) = {𝑦 ∣ ∃𝑥𝐶 𝑥𝐵𝑦}
42, 3elab2g 3665 1 (𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  elima  5927  fvelima  6724  fvelimad  6725  opelco3  32915  fvelima2  41408  afvelima  43243
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