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Theorem elimag 5458
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 4648 . . 3 (𝑦 = 𝐴 → (𝑥𝐵𝑦𝑥𝐵𝐴))
21rexbidv 3048 . 2 (𝑦 = 𝐴 → (∃𝑥𝐶 𝑥𝐵𝑦 ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
3 dfima2 5456 . 2 (𝐵𝐶) = {𝑦 ∣ ∃𝑥𝐶 𝑥𝐵𝑦}
42, 3elab2g 3347 1 (𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wcel 1988  wrex 2910   class class class wbr 4644  cima 5107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117
This theorem is referenced by:  elima  5459  fvelima  6235  opelco3  31652  fvelimad  39274  fvelima2  39291  afvelima  41010
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