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Theorem dfima3 6020
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 6019 . 2 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
2 df-br 5110 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32rexbii 3094 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥𝐵𝑥, 𝑦⟩ ∈ 𝐴)
4 df-rex 3071 . . . 4 (∃𝑥𝐵𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
53, 4bitri 275 . . 3 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
65abbii 2803 . 2 {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
71, 6eqtri 2761 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3070  cop 4596   class class class wbr 5109  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  imadmrn  6027  imassrn  6028  imai  6030  funimaexgOLD  6592  cnvimadfsn  8107  rdglim2  8382  fineqvrep  33760  dfhe3  42139
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