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Theorem dfima3 6072
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfima3 (𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem dfima3
StepHypRef Expression
1 dfima2 6071 . 2 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
2 df-br 5154 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32rexbii 3084 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥𝐵𝑥, 𝑦⟩ ∈ 𝐴)
4 df-rex 3061 . . . 4 (∃𝑥𝐵𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
53, 4bitri 274 . . 3 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
65abbii 2796 . 2 {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
71, 6eqtri 2754 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wrex 3060  cop 4639   class class class wbr 5153  cima 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695
This theorem is referenced by:  imadmrn  6079  imassrn  6080  imai  6083  funimaexgOLD  6646  cnvimadfsn  8186  rdglim2  8462  fineqvrep  34931  dfhe3  43442
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