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Theorem elimag 4897
 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 3943 . . 3 (𝑦 = 𝐴 → (𝑥𝐵𝑦𝑥𝐵𝐴))
21rexbidv 2441 . 2 (𝑦 = 𝐴 → (∃𝑥𝐶 𝑥𝐵𝑦 ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
3 dfima2 4895 . 2 (𝐵𝐶) = {𝑦 ∣ ∃𝑥𝐶 𝑥𝐵𝑦}
42, 3elab2g 2837 1 (𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∃wrex 2419   class class class wbr 3939   “ cima 4554 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-xp 4557  df-cnv 4559  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564 This theorem is referenced by:  elima  4898  fvelima  5485  ecexr  6446
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