ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfima3 Structured version   GIF version

Theorem dfima3 4594
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 4593 . 2 (AB) = {yx B xAy}
2 df-br 3735 . . . . 5 (xAy ↔ ⟨x, y A)
32rexbii 2305 . . . 4 (x B xAyx Bx, y A)
4 df-rex 2286 . . . 4 (x Bx, y Ax(x B x, y A))
53, 4bitri 173 . . 3 (x B xAyx(x B x, y A))
65abbii 2131 . 2 {yx B xAy} = {yx(x B x, y A)}
71, 6eqtri 2038 1 (AB) = {yx(x B x, y A)}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226  wex 1358   wcel 1370  {cab 2004  wrex 2281  cop 3349   class class class wbr 3734  cima 4271
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-cnv 4276  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281
This theorem is referenced by:  imadmrn  4601  imassrn  4602  imai  4604
  Copyright terms: Public domain W3C validator