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Theorem dfima3 4614
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfima3 (AB) = {yx(x B x, y A)}
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem dfima3
StepHypRef Expression
1 dfima2 4613 . 2 (AB) = {yx B xAy}
2 df-br 3756 . . . . 5 (xAy ↔ ⟨x, y A)
32rexbii 2325 . . . 4 (x B xAyx Bx, y A)
4 df-rex 2306 . . . 4 (x Bx, y Ax(x B x, y A))
53, 4bitri 173 . . 3 (x B xAyx(x B x, y A))
65abbii 2150 . 2 {yx B xAy} = {yx(x B x, y A)}
71, 6eqtri 2057 1 (AB) = {yx(x B x, y A)}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wrex 2301  cop 3370   class class class wbr 3755  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:  imadmrn  4621  imassrn  4622  imai  4624
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