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Theorem dfima2 5685
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
dfima2 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem dfima2
StepHypRef Expression
1 df-ima 5325 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 dfrn2 5514 . 2 ran (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦}
3 brres 5607 . . . . . 6 (𝑦 ∈ V → (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐵𝑥𝐴𝑦)))
43elv 3389 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐵𝑥𝐴𝑦))
54exbii 1944 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
6 df-rex 3095 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
75, 6bitr4i 270 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝐵 𝑥𝐴𝑦)
87abbii 2916 . 2 {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦} = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
91, 2, 83eqtri 2825 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wrex 3090  Vcvv 3385   class class class wbr 4843  ran crn 5313  cres 5314  cima 5315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325
This theorem is referenced by:  dfima3  5686  elimag  5687  imasng  5704  dfimafn  6470  isoini  6816  dffin1-5  9498  dfimafnf  29955  ofpreima  29984  dfaimafn  42019
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