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Theorem dfima2 4597
 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 4285 . 2 (AB) = ran (AB)
2 dfrn2 4450 . 2 ran (AB) = {yx x(AB)y}
3 vex 2538 . . . . . . 7 y V
43brres 4545 . . . . . 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 1478 . . . 4 (x x(AB)yx(x B xAy))
8 df-rex 2290 . . . 4 (x B xAyx(x B xAy))
97, 8bitr4i 176 . . 3 (x x(AB)yx B xAy)
109abbii 2135 . 2 {yx x(AB)y} = {yx B xAy}
111, 2, 103eqtri 2046 1 (AB) = {yx B xAy}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008  ∃wrex 2285   class class class wbr 3738  ran crn 4273   ↾ cres 4274   “ cima 4275 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285 This theorem is referenced by:  dfima3  4598  elimag  4599  imasng  4617  imadiflem  4904  imadif  4905  imainlem  4906  imain  4907  funimaexglem  4908  dfimafn  5147  isoini  5382
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