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Theorem dfima2 4613
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) = {yx B xAy}
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem dfima2
StepHypRef Expression
1 df-ima 4301 . 2 (AB) = ran (AB)
2 dfrn2 4466 . 2 ran (AB) = {yx x(AB)y}
3 vex 2554 . . . . . . 7 y V
43brres 4561 . . . . . 6 (x(AB)y ↔ (xAy x B))
5 ancom 253 . . . . . 6 ((xAy x B) ↔ (x B xAy))
64, 5bitri 173 . . . . 5 (x(AB)y ↔ (x B xAy))
76exbii 1493 . . . 4 (x x(AB)yx(x B xAy))
8 df-rex 2306 . . . 4 (x B xAyx(x B xAy))
97, 8bitr4i 176 . . 3 (x x(AB)yx B xAy)
109abbii 2150 . 2 {yx x(AB)y} = {yx B xAy}
111, 2, 103eqtri 2061 1 (AB) = {yx B xAy}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wrex 2301   class class class wbr 3755  ran crn 4289  cres 4290  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  dfima3  4614  elimag  4615  imasng  4633  imadiflem  4921  imadif  4922  imainlem  4923  imain  4924  funimaexglem  4925  dfimafn  5165  isoini  5400
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