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Theorem elimag 5905
 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 5036 . . 3 (𝑦 = 𝐴 → (𝑥𝐵𝑦𝑥𝐵𝐴))
21rexbidv 3221 . 2 (𝑦 = 𝐴 → (∃𝑥𝐶 𝑥𝐵𝑦 ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
3 dfima2 5903 . 2 (𝐵𝐶) = {𝑦 ∣ ∃𝑥𝐶 𝑥𝐵𝑦}
42, 3elab2g 3589 1 (𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   class class class wbr 5032   “ cima 5527 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537 This theorem is referenced by:  elima  5906  fvelima  6719  fvelimad  6720  opelco3  33265  fvelima2  42266  afvelima  44091
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