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Theorem dfima2 5374
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 5041 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 dfrn2 5221 . 2 ran (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦}
3 vex 3175 . . . . . . 7 𝑦 ∈ V
43brres 5310 . . . . . 6 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵))
5 ancom 464 . . . . . 6 ((𝑥𝐴𝑦𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴𝑦))
64, 5bitri 262 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐵𝑥𝐴𝑦))
76exbii 1763 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
8 df-rex 2901 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
97, 8bitr4i 265 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝐵 𝑥𝐴𝑦)
109abbii 2725 . 2 {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦} = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
111, 2, 103eqtri 2635 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wrex 2896   class class class wbr 4577  ran crn 5029  cres 5030  cima 5031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041
This theorem is referenced by:  dfima3  5375  elimag  5376  imasng  5393  dfimafn  6140  isoini  6466  dffin1-5  9070  dfimafnf  28623  ofpreima  28654  dfaimafn  39692
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