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Theorem dfima2 5608
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.)
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 5263 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 dfrn2 5448 . 2 ran (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦}
3 vex 3354 . . . . . . 7 𝑦 ∈ V
43brres 5542 . . . . . 6 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵))
5 ancom 448 . . . . . 6 ((𝑥𝐴𝑦𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴𝑦))
64, 5bitri 264 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐵𝑥𝐴𝑦))
76exbii 1924 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
8 df-rex 3067 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
97, 8bitr4i 267 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝐵 𝑥𝐴𝑦)
109abbii 2888 . 2 {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦} = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
111, 2, 103eqtri 2797 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wrex 3062   class class class wbr 4787  ran crn 5251  cres 5252  cima 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263
This theorem is referenced by:  dfima3  5609  elimag  5610  imasng  5627  dfimafn  6389  isoini  6734  dffin1-5  9416  dfimafnf  29776  ofpreima  29805  dfaimafn  41762
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