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Theorem dfima2 4697
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 4385 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 dfrn2 4550 . 2 ran (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦}
3 vex 2577 . . . . . . 7 𝑦 ∈ V
43brres 4645 . . . . . 6 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵))
5 ancom 257 . . . . . 6 ((𝑥𝐴𝑦𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴𝑦))
64, 5bitri 177 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐵𝑥𝐴𝑦))
76exbii 1512 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
8 df-rex 2329 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
97, 8bitr4i 180 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝐵 𝑥𝐴𝑦)
109abbii 2169 . 2 {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦} = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
111, 2, 103eqtri 2080 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wrex 2324   class class class wbr 3791  ran crn 4373  cres 4374  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  dfima3  4698  elimag  4699  imasng  4717  imadiflem  5005  imadif  5006  imainlem  5007  imain  5008  funimaexglem  5009  dfimafn  5249  isoini  5484
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