ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfima2 GIF version

Theorem dfima2 4721
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 4405 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 dfrn2 4572 . 2 ran (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦}
3 vex 2614 . . . . . . 7 𝑦 ∈ V
43brres 4667 . . . . . 6 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵))
5 ancom 262 . . . . . 6 ((𝑥𝐴𝑦𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴𝑦))
64, 5bitri 182 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐵𝑥𝐴𝑦))
76exbii 1537 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
8 df-rex 2359 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐴𝑦))
97, 8bitr4i 185 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝐵 𝑥𝐴𝑦)
109abbii 2198 . 2 {𝑦 ∣ ∃𝑥 𝑥(𝐴𝐵)𝑦} = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
111, 2, 103eqtri 2107 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wrex 2354   class class class wbr 3806  ran crn 4393  cres 4394  cima 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-cnv 4400  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405
This theorem is referenced by:  dfima3  4722  elimag  4723  imasng  4741  imadiflem  5030  imadif  5031  imainlem  5032  imain  5033  funimaexglem  5034  dfimafn  5276  isoini  5510
  Copyright terms: Public domain W3C validator